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a < bWX=

a ≤ b i a 6= b`

��;VKLO[W;>EBRP=lU�=l«{;VMKÊQsTe:OLU�;LWX=lv[E:=�ANGgRXAZpqOjMPKD;L?BANU2_QMX=�GQsd=�_:=�OD?BA\=lG,?BEQsNsN=lt[GzKD;VRXANKD�VRX=@sN;:`B�+�CADE+�-F. �B` � 0HG>I �Q¥N�´¦V�X¨j¥N�´¨[¥>�l¦�ª^�Q£B¤D¥d�X¦Sª\�6�S�u�B�X�V�l¥}ªKJML,¦C¨[¦H²P¦V�ON}�B�,� ¦��6¦�°��[²QP4¦�¨[¥>�Q£B¤D¥d�P¦�ª\� ¯4¥N�6¥^�B�)¤^¯4¥N¤SRd�X¦¤TP±�,¯±¤L¦ 2 ¥dª\¨[¥P²N¯±� 2 �B� ¤^¯4¥N¤8RN�X¦ m

� �S�<®N�Q¥m ≤ a © �V�X�z²N�Q�S��®d�B¯4¥N�\¤^¯4¥N¤SRd�X¦ a

¨[¥d��²T¦S�ON}�B�,� ¦ 34 ¥N¤D¦�ª}� �P�j²N¯65S� 3 JA{; S = {n :

RPOJ?:OmsNOjGS��E:GgRPOz?BA>GJI:KLANMPO[WZa[EQA>sNOjGgRXACG:v[;nsdOjGgRP=lACG:AkE:G³KL=lACKkf:MPO�KL��G:=lKLO } ` 0 ∈ S_,O[MPagE:AD_,;VMT;�sdEQ;jU�agE:=�ANMoGJI:KLACMXO

ana + 0 = a

;jWX��agE:A0 ≤ a

_6OjM>?BAd],G:=�sN=lt[G2`��e:OjMT;�WXAC;n ∈ S

n<YS;VKLO[Wk;�YjACMkWx==lKL_:U�=@sN;HagE:A

n| ∈ S`�JA{;

CE:G¡sdO[GS��E:GgRXO�sdO[G

n| ∈ C`�¢�E:ANMPANKLOgW�;[WxACvjE:MT;VMFagE:A

CRP=lACG:ADE:G)K>=�ANK^f:MPOmKL��G:=�K>O,`<J=

0 ∈ CANGgRPOjGQsNACWWX��RX=�ANG:A\E:GHKL=�ANKkfQMXOLKL�lGQ=lKLOQ`Q��GgRXOjG,sdACW�WxE:_6OjGQv[;VKLOgW�agE:A

0 /∈ C`Q��GgRPOjGQsNACW�RXOB?BO

a ∈ CACW

b|_Q;VMT;

;VU�vjI:Gby4YjACM�A��ACMPsN=�sN=lO � |}`1U:OjMPKLANKLO[W�ANU�sdOjGV��E:GgRXO

DMPACWxRP;jGQ?BO ¸ ?:A�sN;[?:;\KL=lACKkf:MPOo?BA

CngACW�?BACsN=lM{n

D = {a : a| ∈ C} `

�)�� E� � �

��WsNU�;jMXO>a[EQAn ∈ D

_6OjMTa[EQAn| ∈ C

`B��GgRXO[GQsdA{W�_,O[M�U�;ê:=�_,tjRXACWX=@W�?:AZ=�GQ?BE,sNsd=�tjGDRP=lACG:AFE:GmK>=�ANK^f:MPOkKL�lGQ=lKLO

m`B��O[M

WXE:_:E:A{W�RPO0WxOgWx_6ACsTeQ;jKLO[W�agE:Am| WxA{;^KL=�ANK^f:MXODKL��G:=�K>O0?:A C

`:�r;VMT;>sNOjGB],MXKD;VM��{WxO,nQWxA{;a ∈ C

`Q��GgRXO[GQsdA{Wa = b|

_Q;VMT;;VU�vjIQG

b ∈ D`Q�we:O[MP;

m ≤ bnB_,O[M�RP;jG[RPO�yqYjACMA��ACMPsN=�sN=lO[|

m| ≤ b| = a`�

�� 1 � 14369�2+4*

¸ `�y ¸ | � ANKÊ:ACWxRXMPAFU�;ÛlA i ;jWXOBsd=@;SRP=lYS;^_Q;VMT; +n:agE:AFWx=�ANKL_:MPA

(a + b) + c = a + (b + c)`2yuJAC;

S = {c :_,;VMT;\RXOB?BO

a, bn(a + b) + c = a + (b + c)} i ;j_:Ul=@sN;jM�UlACKL;D�:` ¸ ` |�:`�y �[| � Ad],G:=lACGQ?BO0 × b = 0a| × b = a × b + b?BANKÊ:ACWxRXMPA\U�;0UlA i ?B=@WxRXMP=lf:E:RX=�Y�;�;DU�;0=l«{a[EQ=lACMP?:;Qn a × (b + c) = a × b + a × c

`�JE:v[ANMPANGQsN=�;QhQWxA{;S = {a :_Q;VMT;kRXOB?BO

b, c, a × (b + c) = a × b + a × c} `¹ `�y �[|��r;jMxRP=lACGQ?BOD?BANU<A��ANMTsd=@sd=�OD�BnQ?BACKkE:A{W�RPMXAFU@;>UlA i sdO[G:KkEBRT;SRP=lYS;>_Q;VMT; × `� `�y ¸ |��r;jMxRP=lACGQ?BOD?BANU<A��ANMTsd=@sd=�OD�BnQ?BACKkE:A{W�RPMXAFU@;>UlA i ;jWXOBsd=@;SRX=�YS;k_,;VMT; × `� `�y ¸ | � ANKLOgW�RPMP;jMCnBWX=lGHEQWP;VM_:MP=�GQsd=�_:=�OL?:ANUbf:E:ACGHOjMT?BANG2n:agE:A\_,;VMT;oRPOJ?:O>GJI:KLACMXOa 6= 0

na = b|

_Q;VMT;L;VU�vjI:Gb`

Q`�y ¸ | � ANKLOgW�RPMP;jM�agE:Aa ≤ b ⇔ a| ≤ b|

y4G:OL_:E:A{?BA\EQWP;VM_:MP=lG,sd=�_:=lOD?BAFf:EQANGHOjMT?BANG6|}`�� MXACODa[EQA\�CWxRXOD;VU@sN;jG:«C;>_Q;jMP;^=lG,?B=�sC;VM�sNtjKLOLUlOgW;ScB=�OjKD;jW�?BA\��AC;jG:ODWxO[G�WxEB]6sd=�ANGgRXA{W�_Q;VMT;L?BACWP;VMPMXO[UlU@;VM�U@;kRPANO[MX�@;L?BAGJI:KLANMPO[WC` � AL;jagE:��ANG³;[?BANU@;VGgRPAkEQWP;VMPANKLO[W�KkEQsTe,;jW�OVRXMT;jW�_:MPOj_:=�AC?Q;j?BA{WNn6sdOjKLOz_,O[M�Ax��ANKL_:U�OzU@;jW�UlA i A{W�?BALsN;jGQsdACU�;[sd=�tjG_Q;jMP;>U@;>WXE:KD; i ACU2_:MPOJ?:EQs}RPO i UlA i A{W?BA\AdcB_6OjG:ACGQsd=@;jsN=lt[G2n:Wx=�GH?BANKLO[WxRXMT;VMPU@;jW�?BACWP?BAFU�O[Ww;VcJ=�OjKD;[WNnJ_6ANMPOLWX�<WXAFUlOgW_:E:A{?BA?BACK>OgW�RPMP;jMC`

� A�e:ACsTeQOQnj_6OB?BANKLOgW�?BACKLO[WxRXMT;VMrRPOJ?:O\RXACOjMPANKD;oagE:A�;j_Q;VMPACsNAANG0E:G�Ul=�f:MPOoOoMPANYJ=�WxRP;k?BA�U@;FRXACOjMP��;\?:A�GJI:KLANMPO[WC`j��ANMPOYjACMXACKLO[W�?BACWX_:E:�{WsdO[GzACUbRXACOjMPANKD;^?:A��L?:ANU<agE:AFG:O0WXAF_:E:AC?:Ao?BANKLO[WxRXMT;VM�RPOJ?:OLsNOjGmUlOgW;ScB=�OjKD;jW�?BA\��AC;jG:OQ`

�2+��m1��21 9C1! #"B�+�CADE+�-/.%$A+'&!.($ :�) :�*O:>@A; `<�r;jMP;HRPOB?BO[W\GJI:KLACMXOgW

a, bsdO[G

b 6= 0n<AdcB=@W�RPA0E:G8IQG:=�sNO�_Q;jM

c, r?BAzGgIQK>ACMXOgWFRP;jU�agE:A

a = c × b + r i r < b`

4 ¥d¤D¦�ª}�u�X�j²N¯65S� 3 BAC; b 6= 0`2JAC;

SACU�sNOjGS��EQG[RPOm?:A>U�O[W

aRP;jUlA{WFa[EQA>GQOzANcJ=@WxRXANG³UlOgW

c, rI:GQ=�sNO[WZANGJE:GQsN=�;[?BO[W�_Q;jMP;

a, b`

J=SG:O�A{WwYS;jsN�lOzWxA{;

aWXE�KL=lACKkf:MPO0KL��G:=lKLO,`Q��W

a 6< b_6OjMTa[EQAkWx=

a < bn,ACW

a = 0 × b + a;jWX��agE:A

0, asdEQK>_QUlACG�ANU

ANGJE:G,sd=@;j?BOL_Q;jMP;c, rn,=lGQsNUlEQ=�?BO�U�;LE:GQ=�sN=�?:;[?m_,O[MPagE:A

n × b > aWX=

n 6= 0`:��GgRPOjGQsNACW

a − bACWwE:G�GJI:KLANMPO>KLACG:OjM�agE:A

aagE:AwG:OkACWxRP�\ACGS`j��OjM�RP;jG[RPO\AdcB=@W�RPANG

d, sI:G:=@sdO[W�sdOjG

s < bRT;VU�ACW�agE:A

a− b = d× b+ s`S��GgRXOjG,sdACW

a = (d+1)× b+ s`

�r;VMT;\U@;oE:G:=@sd=@?:;[?bnBWxA{;a = c× b+ r

sdOjGr < b`[��G[RPOjGQsNACW

a− b = (c− 1)× b + r`J��ANMPO

a− b /∈ S_6OjM�a[EQA

a− b < an

ANGgRPOjGQsNACWd, sACW�U@;kIQG:=�sC;LWxO[UlEQsN=lt[G2nB;jWX�2agE:A

c− 1 = dn:OLWxA{;

d + 1 = cn i r = s

nBKLO[WxRXMT;VGQ?:OâgE:Ad + 1, c

A{W�U@;>I:G:=@sN;WXOjU�EQsd=�tjG2`B��ACMXO>ANGgRXO[GQsdA{W

a /∈ S`B��WxRP;DsNOjGgRXMT;j?B=@sNsN=lt[G�WXE:MXv[A\?BAoWxEQ_,O[G:ANMagE:A

SGQODANMT;kYS;jsN�lO,`

�

�<;L?BACK>OgW�RPMP;[sd=�tjGm?BANUbRPANO[MXACKL;LWXE:vj=�ANMPAZUlOgWWx=�vjE:=�ANGgRPACW;VU�vjO[MX=lRXKLOgW�_Q;jMP;>sN;jU�sNE:U@;VMc i rh

cociente(a, b) =9{1

(a < b)� 9{1+"�3

cociente(a − b, b) + 1resto(a, b) =

9{1(a < b) a

9C1!"�3resto(a − b, b)

��; i ?BOgW�sNO[WP;jW<_Q;VMT;wG:OjRP;jM�?BA�ACWxRXOgW�;jUlv[OjMP= RPKLO[WC`CwG,;�A{W�WXEk=�G:Ad]6sd=�ANGQsN=�;>yq_:MPOjf,;VM<ACUBsN�VU@sdEQUlOZ?BA cociente(10000, 7)|}`

, ACK>A{?B=@;VKLO[Wr�{W�RPOo?BACWX_:E:�{WN`g�<;FOjRXMT;FACW�agE:A�Y�;jGLANG0WXEQW�WXEQsdA{Wx=�YS;jWrMPACsNE:MTWx=�OjG:A{W�_,O[M�U�O[WrKL=@WxKLO[W�;VMPvjE:KLACG[RPO[W i Wx=�ANKL_:MPAsdO[GDU�;kKL=�WXKD;o_:MPE:ANf,;a < b

`[��GzKkEQsTe,;jW�WX= RPEQ;jsN=lO[G:ACW�YS;VKLO[W�;kagE:ANMPANM�UlOgW�?BOgW�MPACWXE:UlRP;j?:O[WCnjsdOBsd=�ANGgRPA i MXA{W�RPOQn[ANGgRXO[GQsdA{WsdO[Kkf:=�GQ;jK>OgW�UlOgW?BO[W�ACGmEQGHWXOjU�OL;VU�vjOjMP=lRXKLOQ`B�2OLU�U�;jKD;VKLO[Wcrh

cr(a, b) =9C1(a < b) (0, a)9C1!"�3(c, r) := cr(a − b, b)(c + 1, r)

��ANMPO\G:O[WXOVRPMXOgW�RXMT;VfQ;V�x;VKLO[WrsNOjGLGJI:KLANMPO[W�agE:A�WXAAdcB_:MPACWP;VGDsNOjKLOdk · · ·d0

n[MXAC_:MXA{WxACGgRP;VG,?BOFANUQGJI:KLACMXO k∑

i=0

di×10k

nQagE:AFG:OgW�_,ACMXKL=lRXA\KkEQU RP=l_:U�=@sN;VM_6OjM ¸ �^p4�jsd=�U�K>ACGgRXAj`Q��;jMP;>;V_QMXOSY[ACsTeQ;jM�ACWxRXO,n:sdO[GgYJ=�ANG:AFACU2Wx=�vjE:=�ANGgRPAF;jUlv[OjMP= RPKLOQh

cr10(a, b) =9C1(a < b) (0, a)9C1!"�3(c, r) := cr10(a, 10b)(d, s) := cr(r, b)(10c + d, s)

� � ��

��U�pqE:GQsN=lO[GQ;VKL=�ANGgRXOL?BAFACWxRXA\;VU�vjO[MX=lRXKLOk?:AN_6ANGQ?BA\?BA\agE:AFWx=a = c× 10b + r

sdOjGr < 10b

n i r = db + ssNOjG

s < bn

ANGgRXO[GQsdA{Wa = c× 10b + db + s = (10c + d)b + s

n i ;[?BANKD�[WNnB?B=�MX=�vj=��NG,?BOjG:OgW�;^WXEzAN],sd=�ANG,sd=@;:n d < 10nJ_,O[G:=�ANGQ?BOLU��lKL=lRXA

;VU�RX=�ANKL_6OzagE:A>MXA{a[EQ=lACMXAoACU�sC�VU@sdE:U�O�?BAcr`b�we:O[MP;DGQO0RXACG:ANKLOgW�_:MPOjfQUlACKL;�sdOjG ANU�sdOBsd=�ANGgRXA i MXA{W�RPOz?BA ¸ �j�[�j�Qn��J`6�r;VMT;KD;VGgRXACG:ANM�sdU@;VMPOoANUbsC�VU@sdE:U�OkKL=�ANGgRXMT;jW�UlO>eQ;[sdANKLOgW�;^KL;jG:OQngeQ;[sdANKLOgW�E:G,;oRT;Vf:U@;ksNOjG�EQGQ;\]QU@;^_Q;VMT;osN;[?:;o=�GJYjOBsN;[sd=�tjG�?:A

cr10n:sNOjGmUlOgW;VMPvjE:KLANGgRPO[W i U@;jW�YS;VMP=�;jf:U�ACW�U�OJsC;VU�ACW r, c, d, s :; f s M ? W MPACWXE:U RT;j?BO

10000 7 142 60 8 4 1428, 410000 70 14 200 2 60 142, 6010000 700 1 3000 4 200 14, 20010000 7000 0 10000 1 3000 1, 300010000 70000 0, 10000� tjRXACWXADagE:ADANGÚ@;m]QU�;

in(ci, ri)

ACWoANU�MPACWXE:UlRP;j?:OmACGÚ�;z]QU@;i + 1n(di, si) = cr(ri, bi),

i cibi+1 + ri = 10000n i ACUMXA{WxEQU RT;j?BOL?BA\U@;k],U�;

iA{W

cr(ai, bi)`

��G>ANUBA��ACMPsN=�sN=lO ¹ WXA�_:=@?BA�UlU�ANGQ;jM�U@;�RP;jf:U�;�_Q;jMP;cr10(98713, 11)

`N��;VKLO[W�;ZYjACM�sNtjKLO�e,;jsdACM�A{W�RPOZsNOjG>U�;Z; i EQ?:;�?BA��;V_:U�Ajn_Q;VMT;�ANUjsC;jWXO ¸ �j�[�j��?B=�Yg=@?B=@?BO��B`d��OjG:ACK>OgW�� i eQ;jsNANKLO[W��!�� #"��#��$�� "��%�'&(��$)�*�# eQ;jWxRP;ZagE:A��)+��6`d��GgRPOjGQsNACW�eQ;[sdANKLOgW,�.-��0/�� 21.3'1 �� "54)6)78��#��8"9��# i MPAN_6AdRP=lKLO[W�U�OFWx=�vjE:=�ANGgRPA�e,;jWxRP;Za[EQA��[h��$)�� :1 ��#"�4��'1;3�1 ��!��%�#"4)6)78��!��*�# 9<��#"=7 ��>1�4#?94��#"��@A 1;3)1 ��#"�4)6'7 ��'&�1 �B��!�<��8"278��

U�=�GQ;jUlKLANGgRPA\MXA{sdOjU�ACsdRP;jK>OgW�U@;>RP;jf:U@;^KLA{?B=�;jGgRXAC 3;4ED C 4)3�-F�HG)3I�.-��0/J<�3LK�4�D%M�G#?=��DN�#"��DN� "21��=DN�8"54��=D��#"9<��=D(�#"=7 �=D��#":1;3�1 �=D��#"�4)6)78�=D��@O3�<# �wE:GQagE:A

cr10sNOjGJYJ=lACG:Ak_,;VMT;DUlOgW�agE:A>MPAN_:MPACWXANGgRP;jG�WxE,W�GJI:KLANMPO[W�ANG³ANU�Wx=@W�RPANKD;�?:ACsd=�KD;VUun�G:Oz?BA{Wx_QMXA{sd=�ANKLO[W�;0U�O[W

agE:AF_:MXAN]QANMPANGmOVRPMXOLWX=�WxRXACKL;Qn:?B=�v[;VKLOgW�fQ;jWXABsNOjG

B > 1h

crB(a, b, B) =9C1(a < b) (0, a)9C1!"�3yc, r) := crB(a, Bb, B)

(d, s) := cr(r, b)(Bc + d, s)

��O[M�WXE:_:EQACWxRXOQnJWX=BACW�KkE i vjMT;VG,?BAw_6O[WX=lf:U�ANKLACG[RPAZsNOjGJYJ=lACG:AwEQWP;VM�KLA��OjM�;VU�vjO[MX=lRXKLOoagE:A cr

_Q;VMT;osC;VU@sdE:U@;VMcr(r, b)

`��WxRXO OBsNE:MXMPA³sNEQ;VG,?BO WXA eQ;[sdA³;jMX=lRXKL�NRX=@sN; ?:A vjMT;VG,?BACW�GgIQK>ACMXOgWNh�WXA _:E:A{?BA E,WX;jMCn�_,O[MzAx��ANKL_:U�OQn

B = 216 ACG E:G,;sdO[K>_QEBRP;[?BOjMT;\agE:A�RPMP;jfQ;S�x;\sNOjG ¸ of:=lRPWZy4?:�lv[= RPO[W�f:=lG,;VMP=lOgWP|dng;j_:MXOSY[ACsTeQ;jGQ?BOFU@;k?B=�YJ=�WX=lt[G�?BA�U�;kKL�[agE:=lG,;o_Q;jMP;oWP;jsC;VMd, s`

� ��

�� 1 � 1 3,9 �2+ �¸ ` y ¸ |��;Vv[;

a, b :=MT;VGQ?

(),MT;VGQ?

()KLOB? ¸ �j�[�j�j�[�j� i UlU�ANG:AFU@;>RP;jf:U�;LsNOjKLOÂCGHANU2Ax��ANKL_:U�OQ`�B` yu�j| � ANKÊ:ACWxRXMPA\a[EQA

crB(a, b, B)pqE:GQsN=lO[GQ;:`

¹ ` y ¸ |��2U�ANGQ;jM�U�;>RT;Vf:U@;>_Q;VMT;cr10(98713, 11)

`� ` y ¸ |Z�WP;VG,?BOzU@;H?BAd],G:=�sN=lt[G³_,;VMT;�?BOgW�GJI:KLACMXOgW�G,;SRXEQMP;jUlA{W

a, bagE:A

a ≤ bWx=

b = a + n_,;VMT;�;VU�vjIQG³GJI:KLANMPO

G,;SRXEQMP;jUnnQ?BANKÊ:ACWxRXMPA\agE:A\Wx=

a?:=lYJ=@?BAo;

b 6= 0ACGgRXOjG,sdACW

a ≤ b`

� ` y � |��r;VMT;>sN;j?Q;>;[WxACMPsN=lt[G2nBWx=bA{WYjACMP?Q;j?BACMP;^?BANKÊ:�CWxRXMPANU@; i Wx=bGQODKkE:A{W�RPMXAFE:GHsdO[G[RPMP;jA��ACK>_QUlO,h�[|resto(a + kb, b) = resto(a, b)

ncociente(a + kb, b) = cociente(a, b) + k

`RC|

resto(a + d, b) = resto(a, b) + resto(d, b)ncociente(a + d, b) = cociente(a, b) + cociente(d, b)

`²C|

resto(a + d, b) = resto(resto(a, b) + resto(d, b), b)`

¨:|resto(ad, b) = resto(a, b)resto(d, b)

`¥{|

resto(ad, b) = resto(resto(a, b)resto(d, b), b)`

:` y ¸ |��_:Ul=@agE:A\ANU�;VU�vjOjMP=lRXKLOcr10;>U�O[WWX=lv[E:=lACGgRXACW;jMXv[E:KLANGgRXOgWNh

(300, 7)n(2541, 5)

n(1001, 7)

n(78964, 5)

`�J` y ¸ |m��;VU@sdE:U�A

resto(21378, 7)`Zy � O�ACWzG:ACsNACWP;VMP=lO¡O[fBRXACG:ANM

21378Wx=Z;V_:U�=�sC;8ACU�MPACWXE:UlRP;j?:O8?BA³EQG:O ?BA³ACWxRXO[W

Ax��ANMTsd=@sd=�O[WC` |P ` y��V| � ANKÊ:ACWxRXMPA\a[EQAFRXOB?BOLGJI:KLANMPO>MT;jsN=lO[GQ;VU�_6O[WX= RP=lY[O i KLANG:O[Mwa[EQA ¸ A{W=lv[EQ;VUb;

1

n1+

1

n2+ · · · + 1

nk?:OjGQ?BA1 < n1 < n2 < . . . < nk

`QJE:vjACMXACGQsd=@;:hB��W�RPA\A��ANKL_:U�OLKkE:A{W�RPMP;LsNtjKLOL_:MXOBsNAC?BACMCh7

17=

1

3+

(

7

17− 1

3

)

=1

3+

4

51=

1

3+

1

13+

(

4

51− 1

13

)

=1

3+

1

13+

1

663� =@WxA% :AFE:G�;VU�vjOjMP=lRXKLO^_,;VMT;^sC;VU@sdE:U@;VMU�O[W

nii ;j_:U��agE:ACUlOD;D�0Q ¹ n'�;�0Q ¸ �j�:n ¸ � � ¸ Q ¸ � ¹ �:n �Q ¸�R `

�� <� ��

R `�y ¸ | � ACsN=�?Q;�_,;VMT;�sC;j?:;ZE:GQ;F?BA�U@;jWrWX=lv[E:=�ANGgRXA;jWXANMTsd=�OjGQACW�Wx=BA{W�YjANMT?:;[?BANMT;wOZp4;VU@WP;�_Q;jMP;�U�O[W�ANGgRPANMPO[WCn i ��EQW�RP= ]6a[EQAU�;>MPACWX_:E:A{W�RT;:hB;[|a|a `Bf,| a| − a

`QsC|�WX=a|b ANGgRPOjGQsNACW b|a `Q?,| a|0 `BA�|J= a|b i b|c ACGgRXOjG,sdACW a|c `BpP|B= a|b i b|aANGgRXO[GQsdA{W

a = b`Bvg|J=

a|bc ACGgRXOjG,sdACW�O a|b O a|c `:e6|J= a|b ANGgRXO[GQsdA{W a2|b2 `��

�<+�� 14583697 +�91:<; :>=�:>@A; �B` ��0 �GHGgIQK>ACMXO

nACW © �}¯±¤D¦DWX= n > 1 i WXEQW�I:G:=@sdOgW�p4;js}RPOjMPACW�WXOjG ¸ i n

`�2OgW�_:MX=�KLANMPO[W ¸ �>_:MP=�K>OgWWxO[GH�:n ¹ n � n �Bn ¸[¸ n ¸C¹ n ¸ �Jn ¸%R n�� ¹ n � R `�<OB?BOzGJI:KLANMPO

n > 1ACWZ?B=�YJ=�WX=lfQUlAo_6OjMZ;VU�vjI:G�_QMX=�KLOQ`Q��O[MPagE:A[n,WXAC;

S = {n : n > 1 i G:=�G:vjIQG�_:MP=lKLO0A{Wp4;jsdRXO[M�?BAn} `�J= S GQO0ACW�Y�;[sd��OQn6WxA{; n

WxE�KL=�ANK^f:MXO�K>��G:=�KLOQ`Q��W�RPAnG:O0A{Ww_:MP=�K>O�_,O[MPagE:AoGQODRX=�ANG:Akp4;jsdRXOjM�_:MP=lKLOQ`,��GgRXO[GQsdA{W

nRX=�ANGQAkE:GHp4;[s}RXO[MksNOjG

1 < k < nnn = n′k

`6��OjMwU@;DKL=lGQ=lKD;VU�=@?:;j?�?BAnACG

Snk /∈ S

n i _6OjMwRT;VGgRXO kRP=lACG:AoE:G�p4;js}RPOjM

_:MP=lKLOpnk = k′p

`,��ANMPO0ANGgRXO[GQsdA{Wn = n′k′p � A{W�RPO�?B=@sdAâgE:A n /∈ S

n�sdO[GgRXMT;j?B=@sNsN=lt[GHagE:A^WxE:MPvjA^?BA^WXE:_6OjG:ACM�agE:ASG:O

ACW�YS;[sd��OQ`B�+�C DE+�-/. �B` 0� ��D¯±�{§��Q¯±� ¦�ª © �}¯±¤D¦�ª 34 ¥d¤D¦�ª}�u�X�j²N¯65S� 3 y4��EQsNUl=@?BACWT|}`<BE:_,O[G:v[;jKLO[WZa[EQAkeQ; i ANcB;[s}RT;VKLANGgRXA k

_:MP=lKLO[WCnp1, ..., pk

`2JAC;nANU�_:MPOB?BEQsdRXOz?:AkRPOJ?:O[WCn

n = p1 ×· · · × pk`J��GgRXOjG,sdACW

resto (n + 1, pi) = 1_,;VMT;osN;[?:;\E:G:OL?BA�ACWXO[W

pin[A{W�?BA{sd=�MCngG:=�G:vjIQG

piACW�p4;jsdRXO[M�?BA

n +1`

��ANMPOo;jsC;VfQ;jK>OgW�?BAw?BANKLOgW�RPMP;jM�a[EQA�RXOB?BO\GJI:KLANMPO\KD; i OjMragE:A ¸ RP=lACG:AwE:G>p4;jsdRXO[M�_:MP=lKLOQnj_,O[MrRP;jG[RPO {p1, ...pk}G:O\_:E:A{?BA

WXANMANU2sNOjGS��EQG[RPO0?BAZRXOB?BOgW�UlOgW�_:MX=�KLO[WC`��2OgW�;jUlv[OjMP= RPK>OgW_Q;jMP;D?BA{WXsNE:f:MP=lMZWx=�E:G³GgIQK>ACMXO�ACW�_:MX=�KLO0eQ;jG�OBsdE:_Q;[?BO0U@;�;VRXANG,sd=�tjG�?BA^KL;VRXACKL�VRX=@sdOgW�_6OjMZWx=�vjU�O[WC`

�C��W R;R;R _QMX=�KLOg� � OQn�ACWÂNYJ=@?BANGgRXACKLANGgRXA�?:=lYJ=@Wx=�f:U�A0_,O[M ¹ `��C��W R;R �z_QMX=�KLOg� � O A{Wk?B=�YJ=�WX=�f:UlA�_,O[M ¹ y4U�;³WxE:KD;�?:A�WxE,W?B��vj=lRXOgW�G:OÛlOÂCWT|}ngG:=�_,O[M � n[GQ=6_6OjM �JngG:=6_,O[M ¸[¸ y ;jagE:�6ANKL_:=�AN«NO>;oEQWP;VM�E:GQ;>sN;jU�sNE:U�;[?BOjMT;[|d`;��O[K>ACGgRX=lRXOQhB�C_,O[M�agE:�FWP;VUlRXOU�O[W�GgIQK>ACMXOgWw_Q;jMXA{WP�^��OjMTagE:A R0R �LGQO�ACW�_Q;jMCn i Wx=�pqE:ACMP;z?B=lYJ=@Wx=�f:U�Ak_6OjMZE:G GJI:KLACMXOz_Q;VM{n,WXE³�jMXf6OjU�?BAkp4;[s}RPOjMPACWRPANGQ?:MX�@;�B`��C��O[MâgE:��WP;VUlRXO R ��O[MPagE:ADGQO�ACWk_:MX=�KLOQn i³i ;�e:A0_:MPOjf,;j?BO�WXEQW\p4;[s}RPOjMPACWCn i WX= R pqE:ANMT;mp4;jsdRXO[MCn<RP;VK^f:=��NG¡U�O WxACMX�@;RXOB?BOkp4;jsdRXOjM�?BAZ�NUu`J��GgRXOjG,sdACW�WXtjU�OoYjO i ;o_QMXO[fQ;VM�_:MX=�KLO[W�_Q;VMT;o?B=�YJ=�WX=�f:=lU�=@?:;j?b`BJ=�vjO>sdOjG ¸C¹ n ¸ �Bn ¸�R nB� ¹ nJ� R n ¹Q¸ ` � =�G:vjEQG:OACWZp4;[s}RPOjM\?BA R;R �J`2��J=lv[OmsNOjG ¹ �j� � OQ`b��O[MPagE:A

372 > 997n i Wx=�pqE:ACMP; 997 = 37x

RXACGQ?BMP��;magE:ADWxACMx < 37 � Wx=rG:O,n2Wx=

x ≥ 37n37x ≥ 372 agE:ALA{WFKL; i OjMZa[EQA R0R �B`,��Wk?BACsN=lM{nbWx=reQ; i EQG p4;[s}RXO[MFKL; i OjM�Om=�vjE,;VUragE:A ¹ �Jn�eQ;jf:MP�zOVRPMXOmKLANG:O[MagE:A ¹ �Jn6_6ANMPO i ;zWx�LagE:AkGQOzeQ; i _6OjMTagE:Ake:A>_:MPOjfQ;[?BO0RPOB?BO[W�UlOgW�_QMX=�KLO[W�K>ACG:OjMPACWZagE:A ¹ �J`6��GgRXO[GQsdA{WZ_,OB?BACK>OgW�?:ACsd=�M{h_Q;jMP;z_:MPOjfQ;jMFWX=rE:G)GJI:KLANMPO

nACWF_:MP=�K>OHfQ;[W�RT;�_QMXO[fQ;VMZRXOB?BOH_:MP=�K>O

pRP;jU�a[EQA

p2 ≤ n`<��;jMP;zGJI:KLANMPO[WZKLANG:O[MXA{WFa[EQA

¸ �[�j�zACWxRXOHKLA0Wx=�MPYjALf:=�ANG´_,O[MPagE:ALG:OHKLADOjfQUl=�v[;H;m=lM\KD�[Wo;VU�U��HagE:A ¹:¸ n i WX�DsN;jWX=�?BADKLANKLO[MX=@;mUlOgW ¸j¸ _:MP=lKLO[WFe,;jWxRP;¹:¸ hJ�:n ¹ n � n �Bn ¸[¸ n ¸C¹ n ¸ �Jn ¸%R n�� ¹ n � R n ¹Q¸ `SJ=bWxE:_Q=lACMP;^?BA�KLACK>O[MX=@;oU�O[W�_:MP=lKLOgW�eQ;[W�RT; ¸ �j�\KLAFWxACMXYJ=�MX�@;VG2ngsdO[GzsN;jU�sNE:U@;j?BO[MP;QnV_Q;VMT;_:MPOjfQ;jMCn�ACG)U@;mKL=�WXKD;�pqO[MXKD;QnbU�;m_:MP=lKD;jUl=@?:;j?)?BALGJI:KLANMPO[WFeQ;[W�RT; ¸ �zK>=�Uu`�J=�G:A{sdA{Wx=lRXOH_:MPOjfQ;jMFGgIQK>ACMXOgWF?BANU�O[MP?:ANG´?BA¸ �[�j�[�j�:ngRPANGQ?BMP�@;0agE:AkWP;Vf6ANM�?BAkK>ACKLOjMP=�;LU�O[W_QMX=�KLO[WwKLACG:OjMPACWagE:A ¹Q¸ Q`Q��; i � y4UlO0sC;VU@sdE:U��ksdO[G�U�;>pqEQGQsd=�tjG�=lRXeQ_:MX=�KLA?BA ��;j_:U�Awa[EQA�?:;FANU

i� �CWX=�K>OF_:MP=�K>OJ|}`[B=QG:OoWX�ACWxRXOgW�?BAKLANKLOjMP=@;�ACGgRXOjG,sdACW�YjO i ;FEQWP;VMrKD�[W�RX=�ANKL_6O\YjANMP=l],sN;jGQ?BO\Wx= ¸ �;�Bn_6OjMA��ACKL_:UlO,nQA{W�_:MX=�KLOQnQagE:AoWX=�K>_QUlACK>ACGgRXAo?B=�YJ=�?B=�ANG,?BOD_,O[M�NUu`:��ANMPOLU�;[WwsdO[K>_QEBRP;[?BOjMT;jW�WXOjGmf:E:ANG,;jW_Q;jMP;>WP;Vf6ANMwsdOgWX;[W

?BAwKLACK>O[MX=@;:` � OoA{W�GQ;[?:;\sN;jMXvg;VMrE:G,;\KD�jagE:=�GQ;\sdO[GDUlOgW�_:MP=lKLACMXOgW � _:MP=�K>OgWNn i EQGD_:MXO[vjMT;VKD;FagE:Aw_:MPE:ANf,;\_:MP=lKD;VU�=@?:;j??BAFE:GzGgIQK>ACMXO>KLANG:O[Ma[EQA ¸ �j�j�[�j�F_:MPOjf,;VGQ?BOÛlOgW � _:MX=�KLO[W�E:G:O>RXMT;jW�OVRPMXO,`:J=�eQ;Vf:U@;VKLOgW�?:A�_:MP=�K>OgW�K>ACG:OjMPACW�agE:AFE:GKL=lU�U�tjG2nbGJE:A{W�RPMXO�;VKL=�vjO R;R �DA{WF_:MX=�KLOHGgIQK>ACMXO ¸ P nb;jWX��agE:ALU�;m=@?BAC;H?BA0sC;VMPv[;jM�E:G,;�RT;Vf:U@;H?BA

n_:MP=�K>OgWF_Q;VMT;z_,OB?BACM

?BA{sd=@?B=lM_QMX=�KD;VU�=�?:;[?�e,;jWxRP;n2 G:OLACWxRP�^RP;jGHKL;jU `��ANMPOweQ; i E:GQ;wsNOjGQWxRP;jG[RPA�sdO[KL_,ANRXANG,sd=@;w_Q;jMP;�Y[ANM<agE:=�ANGo_QE:AC?BA�ANGQsNOjGgRXMT;VM<ANUj_QMX=�KLOwKD�jW�vjMT;VGQ?BA[`d��UJ;jsdRXEQ;jU[sN;jKL_,ACtjGU�UlACGQ;VMP��;zE:GÚ�=lf:MPOHvjO[MP?BOHsdO[G)?:�lv[= RPO[WCn i ;HagE:ALANU�� ¹ ?:AL��vjOgW�RPOH?:AD�j�j� P pqE:A0?:ACWPsdE:f:=�ANMXRXOHACG)F��<� sdO[G ¸ �B` R � P ` R;P0Rsd=lpqMT;jW�?BA{sd=�KD;VU�ACWC`j��W

2p −1njsdO[G

p_:MP=lKLOoA�=lv[EQ;VU6; � ¹ ` ¸[¸ �B` j� R � ��;jWxRP;\;VeQOjMT;FG:OoeQ; i K>�NRXOB?BO^vjANGQANMT;VUB_Q;jMP;\_QMXO[fQ;VMrU@;_:MP=lKD;jUl=@?:;j?\?:A�sdE,;VU@a[EQ=lACM2GJI:KLANMPO�agE:A�WXAC;WX=lv[G:=l],sN;VRX=�Y�;jKLANGgRXA�KLAx��OjM<ACGkWXANGgRX=@?BOw_:MT�jsdRX=@sdO�agE:A�ANUJa[EQA�e:ACK>OgW2ACGgEQGQsd=@;j?BO,n

ACW�?BA{sd=�MCnB_QMXO[fQ;VM�sN;[?:;o_6O[WX=lfQUlA\?B=�YJ=�WXOjM�_:MP=lKLO>K>ACG:OjMagE:AFU@;kMT;V��«ZsNEQ;j?:MP;[?:;:`g��UbGJE:ANY[O^;jUlv[OjMP= RPK>O>�F&o�n:;jUba[EQAZYS;jK>OgW;D?:;jME:G�YJ=�WxRP;j«NOLKD�jWw;[?BANU@;VGgRXA[n:v[;jGQ;>ANG�sd=�ANMXRXO0WXANGgRP=�?BOLRPANt[MX=@sdO,nB_,ACMXOLeQ;[W�RT;D;Ve:O[MP;^G:O0_:MT�jsdRX=@sdO,`:��; i KL�dRPOB?BO[WKD�[WMT�V_:=@?BO[W�_Q;VMT;F?BACsN=�?B=�M�sNOjGL_6OJsC;\_:MPOjfQ;jf:=�Ul=@?:;j?L?BA�ANMPMXO[M�WX=QEQGDGgIQK>ACMXO\A{W�_:MP=�K>O,n i ACW�=�GgRXACMXA{WX;jG[RPA�G:OVRT;VM�agE:AwU@;FpqE:G,sd=�tjG¯@ª © �}¯±¤D¥ ?:A � ;V_:U�AzE,WX;³E:G:O¡?BAmACWxRXOgWmy type(x, prime)

|dn�a[EQAm?B=@sdA �d�V��ªN¥�sdE,;VGQ?BOxG:O)ACW>GQ;SRPE:MT;VUun�=�GJYjOBsN; ¯@ª © �}¯±¤D¥sdE,;VGQ?BO UlO³ACWCh�G:O)?B=@sdA�G,;j?:;³;VU�MPACWX_6ACs}RPO³sNOjG��¥X¶[� © �}¯±¤D¥dn�_,ACMXOO��;V_:U�Az?:ANKLOjMT;�KD�jWÂNG nextprime(10500)

agE:AzANG_:MPOjfQ;jM�U�O[WZ=�KL_Q;VMPACW�KD; i OjMFagE:A 10500

sNOjGisprime

eQ;[W�RT;0ACGQsdO[G[RPMP;jM�EQG:OQÈU:O[MXKD;jUlKLANGgRPAjn�ANUr;VU�vjO[MX=lRXKLOzKL�[WFWx=�KL_:UlA_Q;jMP;>?BAdRPANMPKL=lGQ;jM_:MP=lKD;VU�=@?:;j?zACWCh

primo(n) = primo(n, 2)� 3#" � �primo(n, d) =9{1

(d2 > n)9��

9{1+"�3´9C1(resto(n, d) = 0)

"�3�9{1+"�3primo(n, d + 1)

��! #"$��%�"$�� &��')(��'+*,��-��.$��0/��%�1*��2��"3�,�546��'+�7�2p

− 1

�$��%p

�'+*��!��8�"$��"$�$*�'�8�9#:<;>=@?�A5BDC5EFCG:<A<C�H�H&CGI�JK�MLM"p

"$��"$�%�N��!"$'M�3ODPQ��"��R �'+*��!��K��"0ST"$'+�1"$%�%�"0�$��%��U�$*��8DVW�$��%@"$�1�5XZY\[^]`_D��%�acb�%@ �'+"��!*��Z��"3d�e�e�I e�e�e5��,��'M"$�\ &��'R�1"$'f"$�� '+*��!"$'M�3"$%@��"��M�$b�g�'+*�'b�% �'+*��!�W��"Q�7h��)��"Wd�eW�!*��%�"$�\��"3��i�_�* L1��I�JR�j%�N��!"$'M�2ODk�8��$��%

p = 35,156, 667

8U4lb�"Q��"$�1�$b�g�*�"$'mL1�nc�1"$�7��%��)��"$�1 �b�.$�0V7��*�ad�d�I�d�o�P�I nDp�n��i�_�* LM��G8� #"$'+��*�"$%��7"$�j �'+"$�!*��8�q�b�"3"$�MLMb�(D�7��g�*�"$'mL1�!��1*jb�%��W��.$��I

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��;jU�YjAN«FACU6U�ACsdRXO[M�?:AdRXA{s}RPA\E:GQ;^=lG:AN],sd=�ANG,sd=@;ÂCGz�{W�RPOQhJE:GQ;>Y[AN«Fa[EQAFWP;VU�AZG:OD?B=lYJ=@Wx=�f:U�A�_6OjM��:n:�N_,;VMT;âgE:�FYS;VKLO[W�;^_:MPOjfQ;jMUlOgW�sN;VG,?B=�?Q;SRXOgW�_Q;VMPACWT� � AdRXACMXKL=�GQ;VKLOgWWx=bA{W�=lKL_Q;jM_:MX=�KLANMPOQn i _:MPOjf,;VKLO[W�WXOjU@;VKLANGgRXAF?B=�YJ=�WXOjMPACW�=�K>_,;VMPACWCn i ANU<sC�VU@sdE:U�O?BANf6ANMP�@;^MPAC?:EQsd=�MPWXAZANGHU�;>KL=lRP;j?mKD�jW�OLKLANGQO[WC` ~ GgYJ=lRP;jK>OgW;VUbU�ACsdRXOjMw;>A{WXsNMX=�f:=�M�ACU<;VU�vjO[MX=lRXKLOL;jWX� `�rOjU�Yg=�ANG,?BO>;jU6RXANKD;>?BAZANKL_:U�AC;VM�E:GmsdO[GS��E:GgRXOL?BA

n_:MP=lKLOgWNnB�{sdOjKLO>ANG:v[ANGQ?:MP;jK>OgWrRT;VUbsdO[GS��E:GgRXOJ�\wGQ;^_6O[WX=lf:=�U�=�?:;[?

WxACMX�@;ÂCWxRXAo;VU�vjO[MX=lRXKLO © �}¯±¤D¦�ª � �SªT� �za[EQAFsNOjGgRXA{W�RT;>ANU2sdO[GS��E:GgRXOD?BAFRPOB?BO[W�U�O[W�_:MP=�K>OgWagE:AFG:ODAdc:sdA{?BANGHWXEm;jMXv[E:KLANGgRXO,h

primosHasta(n) = P (n, {}, 2)� 3 " � �P (n, C, d) =9{1

(d > n) C9{1+"�3´9C1(primo(d))P (n, C ∪ {d}, d + 1)9{1+"�3

P (n, C, d + 1)

��OjKLODANG�ACU�sC;jWXO0?BA © �}¯±¤L¦Vn6WXAo_:E:A{?BAoKLA��O[MP;jMwACWxRXODf,;jWxRP;VGgRPAo_,O[G:=�ANGQ?BOm�>ANG�ANU�sdO[GS��E:GgRXOD_QMX=�KLANMPO i YJ=@Wx=lRP;jGQ?BO�WXtjU�OGJI:KLANMPO[W=�KL_Q;VMPACWw?:ACWX_:E:�CWC`Q��ACMXOD_Q;jMP;L_QMX=�KLO[Wwe,;jWxRP;nE:G�RT;VU<��MP;VRXt[WxRXACG:ACWRXE:Y[O0E:GQ;D=@?BA{;DvjACG:=�;jU `Q��KL_,AC«C;VKLOgW�sdO[G

ANU�sdOjGV��E:GgRXO {2, ..., n} KD;VMTsN;jGQ?BOHANU�_:MX=�KLANM^GgIQK>ACMXO,n��Bn�sdO[K>O�_QMX=�KLO i f6OjMPMP;jGQ?BOmRXOB?BOgWoU�O[W\OjRXMPO[W\KÎ:UlRX=�_:UlOgWk?BAm�Bn?BA��x;jGQ?BOL�CWxRXO,nB_Q;VMT;ÂNU<sC;jWXOn = 31

h2 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31?BACWxRP;[sN;jGQ?BO�ACU��0_Q;jMP;0=lGQ?:=�sC;VM\agE:AÂ{WZ_:MX=�KLOQ`��we:O[MP;�ANUr_:MP=�K>ACM�GJI:KLACMXOzG:OzKD;jMPsC;j?BOm?BACWX_:E:�{W�?BA0�:n,ACU ¹ n�ACWZ_:MP=lKLOQ`��GgRXO[GQsdA{W�UlOLKD;VMTsN;jK>OgW i f,O[MXMT;VKLOgW�RPOB?BO[WWXEQW�OVRPMXOgW�KkI:UlRX=�_:U�O[WCh2 3 5 7 11 13 17 19 23 25 29 31� A\GJE:ACYjO,ngACUb_:MP=lKLANMGJI:KLACMXO>G:ODKD;VMTsN;[?BOkA{W�_:MX=�KLOQnBACG[RPOjGQsNACW�U�OLKL;jMPsC;VKLO[W i f6OjMPMP;jKLO[W�RXOB?BOgWWxEQW�OjRXMPO[W�KkIQU RP=l_:U�O[Wy4WXOjU�OLANU<� � |}h2 3 5 7 11 13 17 19 23 29 31� AZGJE:ANY[OkACUb_:MX=�KLANM�GJI:KLANMPOkG:O>KD;VMTsN;j?:OoACW�_:MP=lKLO,ng_6ANMPOL;j?BACKL�[W�WXEzsNEQ;j?:MP;[?BOQn � R nJA{W�KD; i O[M�agE:A n = 31

ngANGgRPOjGQsNACWG:ODagE:AC?:;>GQ;[?:;>KD�jWagE:A\_:MP=lKLOgWN`J=�pqE:ANMT;

n = 190YjACMX�@;VKLOgWNn<sC;VK^f:=�;jGQ?BO�U@;�GQOVRP;[sd=�tjG¡_Q;jMP; a[EQAmagE:AC_Q;�ANG�U�; _Q�Vv[=lG,;:n<ACKL_,AC«C;VG,?BO³sNOjG8U�O³agE:A

agE:AC?:;L?BA{Wx_QE:�CW?BA\f6OjMPMT;VM�KkI:UlRX=�_:U�O[Ww?BAk�:n ¹ i � h��

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i ACWXO[WWXOjGzRXOB?BOgW�UlOgW�_:MP=lKLO[W�e,;jWxRP; ¸%R �QnB_,O[MPagE:AFG:OLeQ;jf:MT�kKÎ:U RP=l_QUlOgWw?BA ¸ �k_:E:A{W�RPODa[EQA 172 > 190`

��W�RPA�_:MXOBsNAC?B=�KL=lACG[RPOZWXA�UlU@;VKD;�U�;L²N�}¯6RP�>¨j¥��X�V�K5Sª}� ¥N�6¥dª}`V�2OF?BAd]QGQ=lKLO[W�pqO[MXKD;VU�KLANGgRXA�ANG^RX�CMXKL=�G:O[W�?BAcriba(P, C, n)sdE i O[W�RXMPACW�;VMPvjE:KLANGgRPO[W�WxO[G2hBE:G�sdO[GS��E:GgRXO P

?BA\_:MP=lKLOgW,E:G�sNOjGS��E:GgRPO

C?BAFGJI:KLANMPO[W ≤ n i ANUb]QG,;VU<?BA\U@;>f:IQWPagE:AC?:;

nh

criba(n) = criba({}, {2, ..., n}, n)� 3 " � �criba(P, C, n) =

d :=ACUbANU�ANKLANGgRPODK>��G:=�KLOL?:A

C9{1(d2 > n) P ∪ C9{1+"�3

criba(P ∪ {d}, C− RXOB?BO[W�U�O[W�KÎ:UlRX=�_:UlOgW?BA dnn)

��OjGHsN;[?:;>=lGJYjOBsC;jsd=�tjGz?BAcriba(P, C, n)

WXA\sdE:KL_:U�ANGHU@;jWWX=lv[E:=�ANGgRXA{WNh��U2ANU�ANKLANGgRPODK>��G:=�KLO

d?BA

CA{W�_:MX=�KLO

PACW�ACU<sdO[GS��E:GgRXOD?BAFRPOB?BO[W�U�O[W�_:MP=�K>OgW�K>ACG:OjMPACWagE:A

d`

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criba(n)`VJE:_6OjGQv[;wagE:A�WxA�sdE:KL_:U�A�_,;VMT;w;VU�vjE:G,;=lGJY[OJsC;jsN=lt[G

agE:A\ANG:v[ANGQ?:MP;ÔVRPMP;Q`QJAC;C ′ = C− KkI:UlRX=�_:U�O[Ww?BA d

`6JA{;eANU<ANU�ANKLANGgRXOLKL��G:=lKLO0?BA

C ′ `Q�wU�vjI:GH_:MP=�K>O p > dA{W�p4;js}RPOjM

?BAe`6��ANMPO

p ∈ C i p ≤ en�_6OjM�RP;VGgRPO

p = e,KLO[WxRXMT;VGQ?:OzagE:AkACU�GgEQANYjOm;VMPvjE:KLACG[RPO�sdEQK>_QUlA>U�;z_:MP=lKLACMP;�_:MXO[_:=�AC?:;[?b`

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d < p < e`B��ANMPOLWX=2eJE:f:=�ANMT;^RP;jUb_:MX=�KLO

pnpG:ODACW�KÎ:UlRX=�_:UlO0?:A\G:=lGQvjI:GHANU�ANKLACG[RPOD?BA

Pn i p ≤ n

_6OjMTa[EQAp < e

n i�i ;DagE:AoG:OD_:EQAC?BAkWXANM�KkI:UlRX=�_:U�O�?BA enp ∈ C ′ nQ_6ANMPODACWxRXO0sdO[GgRXMT;j?B=@sdAoagE:A e

WxA{;LANU�ANKLANGgRPODK>��G:=�KLO�?BAC ′ `,��;

�� <� �� ?� �� %�E� ��

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?BA\ACWXOQnBeQ;jf:=�ANGQ?BOLACK>_6AN«{;j?BODsNOjG {2, ..., n} `��

�� 1 � 14369�2+l�¸ `�y �[|��OJ?:= ],agE:A\ACU2;VU�vjO[MX=lRXKLO © �}¯±¤D¦L_Q;VMT;LagE:AFANYJ= RPA\_:MPOjfQ;jM�GJI:KLACMXOgW�_,;VMPACWCnJWP;VU�YjOD�B`�:`�y � |��MXO[vjMT;VKLAL²N�}¯6RT�DACG � ;V_:U�Aj`��6�[¥N�X¥N�6²N¯4�ShQ��;Vv[;ÂNG ��;j_:U�A

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¹ `�y � | � =@WxA% :AoE:G³;VU�vjOjMP=lRXKLO�y4WX=lG�_:MPOjv[MP;jKD;VMPUlOÂNG ��;j_:UlA�|agE:A\ANKL_:U�AC;0E:GQ;DU�=�WxRP;D?:A\RXOB?BOgWUlOgWw_:MP=�K>OgWKLANG:O[MagE:A ¸ �j�j�QnV_Q;jMP;oWP;jsC;VMrRPOB?BO[W�UlOgW�_:MP=�K>OgW�ACG[RPMXA

n i n + kng?BOjGQ?:A ¸ �[�j� ≤ n i n + k < 1000000

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nKLOB?

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P `�y �[|��GQsNE:ANGgRPMXAÊ:G KL�dRPOB?BO�_Q;jMP;0ANGQsNOjGgRXMT;VMnGgIQK>ACMXOgW�sNOjKL_:E:A{W�RPO[WZsdOjG,WxA{sdEBRP=lY[O[WC`,JE:v[ANMPANGQsN=�;Qh��OjKL=�ANGQsNA

;>f:EQWPsN;VMsNOjGn!`

R `�y �[|��;VU@sdEQUlAn2 + n + 41

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¸ �Q`�y ¸ |��Ulv[E:=�ANGHKLAo?B= ��ODagE:A\U@;^pqE:GQsd=�tjG�?BAN]QG:=@?:;L;jWX� hf(n) = nt(n) + 2(1 − t(n))� 3 " � �

t(n) = resto(M(n − 1, n)2, n) �M(n, m) =

9C1(n = 0) 1

9C1!"�3resto(nM(n − 1, m), m)ANG:v[ANGQ?:MP;jfQ;wWXtjU�O�_:MP=lKLOgWNn i RXOB?BOgW<U�O[W�_:MP=lKLOgWN`{��MPE:�Nf6ANU@;�_,;VMT;wE:G:OgW�sNEQ;VGgRPO[W�Y�;jUlO[MXA{W�?:A n > 0

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a i bACWACU2KD�ScB=lKLO0?B=�YJ=�WXOjM

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mcd(a, b) =d := mın(a, b);

9C1(d = 0) max(a, b)

9{1+"�3mcd(a, b, d)� 3#" � �

mcd(a, b, d) =9{1

(resto(a, d) = 0 � (resto(b, d) = 0) d9{1+"�3

mcd(a, b, d − 1)

U�OâgE:A�?:=�sNA�_:MPOjf,;VM�RPOB?BO[W�UlOgW�_6O[WX=lfQUlA{W�p4;[s}RPOjMPACW�sNOjKÊ:G:ACW�?BAa i b

?:ACWP?BAmın(a, b)

eQ;jWxRP; ¸ i ;jsNAN_BRT;VM�ANU�_:MP=lKLANMPOkagE:AWXAZACGQsdEQANGgRXMT;:`B��ANMPOL��EQsNUl=@?BACWweQ;jsNAZKD�[W?BAo?BO[W�KL=�UlACG:=�O[W?BA{WXsNE:f:MP=lt>E:GHKL�dRXOB?BODKD�[W�MP�j_:=�?:OQh

��U�;VU�vjO[MX=lRXKLO^?:A\��EQsdU�=@?BACWmcd(a, b) =

9{1(b = 0) a

9{1+"�3mcd(b, resto(a, b))

��

JAw_:EQAC?BA�_Q;VU�_Q;VM�ANU,pqE:GQsd=�OjG,;VKL=lACGgRXOk?BA��CWxRXOksNOjG ��;j_:UlA�_,O[G:=lACGQ?BOÛlOgW�?BO[W�;VMPvjE:KLACG[RPO[WrACGa, b i MXAC_:= RP=lACGQ?BOoKL=�ANGgRXMT;jW

b 6= 0a, b := b, a

KLOB?b�<;mMT;V«Nt[G _6OjM\U@;Ha[EQADANU�;jUlv[OjMP= RPK>Om?BAD��EQsNUl=@?BA{WFpqE:GQsN=lO[GQ;mACWoagE:ALU�O[WZp4;jsdRXO[MXA{W\sdOjKÊ:G:A{WZ?:A

a i bWxO[G³ANc:;js}RT;VKLANGgRPA

UlOgWwKL=@WxKLO[W�agE:AoU�O[W�?:Ab i resto(a, b)

`,��O[MCn6WxA{;a = cb + r

sNOjGr < b

`6J=dA{Wwp4;[s}RPOjM�sNOjKkIQG ?BA

a i bna = md i

b = ndnBACGgRXOjG,sdACW

md = cnd + rODWXAC;

r = (m − cn)dnQ;[Wx�bagE:A

dACW�p4;[s}RPOjMwsdO[KkI:Gm?BA

b i r`B� WX=

dACW�p4;[s}RPOjMsdO[KkI:G

?BAb i rnb = nd i r = md

n:ACGgRXOjG,sdACWa = cnd + md = (cn + m)d

nQ;jWX�<agE:AdA{W�p4;jsdRXO[MwsdOjKÎ:G�?BA

a i b`,J=�UlOgW?BO[W

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E:GQ;0?:Ad]QG:=@sd=�tjG�agE:AkG:O0sdO[MXMPACWX_,O[GQ?BAo;DWXE�G:OjK^f:MPAj`Q�agE:��UlODRPANG:ACKLO[W�sNOjKLOLRXACOjMPANKD;:hQ��U�GJI:KLACMXO0sNOjGHACWP;D_:MPOj_:=�AC?Q;j?RX=�ANG:A\agE:A\WXANM�ANU2KD�VcJ=�KLO>sNOjKÎ:Gz?B=�YJ=�WXOjM�_6OjMACUbA��ANMTsd=@sd=�O ¹ ` , A{sd��_:MXOBsC;VKLANGgRXA[nBWxA{;

g = mcd(a, b)n i WXAC; d

E:G�?B=lYJ=@WxO[MsdO[KkI:G ?:A

a, b`¡��MXOBsdA{?BANKLOgW�_6OjMZ=lG,?BEQsNsN=lt[G³WXOjf:MPA

b`bB=

b = 0ACG[RPOjGQsNACW

g = a i dA{W�?B=�YJ=�WXOjMF?BA

a = g.JA{;

bRT;VU

agE:Ao_Q;jMP;>RPOJ?:Ob′ < b i RPOB?BO a′ n,sNEQ;VU@agE:=lACM�?B=�YJ=�WXOjM�sdOjKÎ:G�?BA a′, b′

ACW�?B=�Yg=@WXOjM�?BAmcd(a′, b′)

`6��ANKLO[W�Yg=@WxRXO�agE:AoU�O[W?B=�Yg=@WXOjMPACW�sdO[KkE:G:A{W?BA

a i bWXOjGmAdc:;jsdRP;jK>ACGgRXAFUlOgW?BA

b i r = resto(a, b) < b`B��O[MACWXOQnBU@;Le:=l_6tVRPACWX=�W?:A\=lGQ?:EQsNsN=lt[G2n iANUbe:A{sTe:ODagE:A\e:ANKLOgW�Yg=@WxRXODagE:A

mcd(a, b) = mcd(b, r)n,sNEQ;VU@agE:=�ANM?B=�Yg=@WXOjMwsdO[KkI:GH?BA

a, b?B=�Yg=@?BA\;

g`

� ��

�� 1 � 1 3,9 �2+ �¸ ` y ¸ |��r;VU�_,A)ANU\;VU�vjO[MX=lRXKLO ?:A ��EQsNUl=@?BA{W�sdOjG ��;V_:U�A³sNOjKLO�WXA´?B= ��O MPACsd=�ANGgRPANKLANGgRXA[n�_6OjGQ=lACGQ?BO

�WXANA{?bh��F� P � �

a, b :=MT;VGQ?

(),MP;jGQ?

()y4ANG ��;j_:UlA[n

aKLOB?

b?:;>ANUbMPACWxRXOD?BA

a?:=lYJ=@?B=�?:O

b|}`

�B` yu�j|��O[G ��;V_:U�Ajn,sdOjGgRT;VM_Q;jMP;>KL=lU�_Q;VMPACW?:AoGJI:KLANMPO[W;VU�;V«C;jMoyqMT;VGQ?<y |x|dn,sNEQ�VGgRXOgW�RX=�ANG:ACG�WxEHKDsN?�� ¸ n i sNEQ�VUA{WANUbKD�VcJ=�KLOD?BA\UlOgW�KL=lU2KDsC? i AdcB_:MPACWXA�A{WxA\I:UlRX=�KLODsdOjKLOL_:MPOB?BEQs}RPOD?BA\_:MP=lKLO[WC`¹ ` y ¸ | � ACKkE:A{W�RPMXALagE:A>E:G³?B=�YJ=�WXOjMFsdO[KkI:G)?BAa i b

agE:AÂ{W�?:=lYJ=@Wx=�f:U�A^_6OjMZRXOB?BOH?B=�YJ=�WXOjMZsdOjKÎ:G³?:Aa i b

A{WZANUKD�VcJ=�KLODsdOjKÎ:GH?B=lYJ=@WxO[Mw?BA

a i b`

� ` y ¸ | � ANKÊ:ACWxRXMPA\a[EQAmcd(ka, kb) = k × mcd(a, b) �� ` y � |��WxA\ACUbA��ANMTsd=@sd=�OD;VGgRXACMX=�OjM�_Q;jMP;>?BACK>OgW�RPMP;jMa[EQAZACU<WX=lv[E:=lACGgRXA\;VU�vjO[MX=lRXKLOLsN;VU@sdEQU�;

mcd(a, b) :mcd2(a, b) =9{1

(a = 0) b9C1!"�3)9C1

(b = 0) a9{1+"�3

9 � 7ca, ra = cr(a, 2); cb, rb = cr(b, 2)

yq_6OjMwsdE,;VU@a[EQ=lACMw;VU�vjOjMP=lRXKLOg|9{1

(ra = 0)9C1(rb = 0)

� ×mcd2(ca, cb)9C1!"�3mcd2(ca, b)9{1+"�3

9C1(rb = 0) mcd2(a, cb)9C1!"�3

mcd2(max (a, b) − mın(a, b), mın(a, b)):` y ¸ |��;VU@sdEQU�;jMmcd(a, b)

;kKD;jG:O>WX=�vjE:=�ANGQ?:OÂCU2;VU�vjOjMP=lRXKLO^?BAZ��EQsNUl=@?BA{W_Q;VMT;(a, b) =

y ¸ �Jn ¹ |}n�y ¹ n ¸ �V|dn�y ¸ � ¸ n��j�j|dny R;R0R n R;R |dnby ¸ �[� � n ¸ � |}n�y R0P � ¸ n � R � |}n�y ¹ �;�Jn�� ¹[¹ |d`

�J` y ¸ | � ANKLO[WxRXMT;VMagE:AoWX=pACW�_:MP=�K>O,n

m > 0n i pG:OLACW�p4;[s}RXO[Mw?BA

bnBACG[RPOjGQsNACW

mcd(pm, b) = mcd(m, b)`

� ��

�2+ �� 3 �� 587� � " � 7:5 � "��{7 � � �� 7)- � 1��C5��C1 � 7J=<E:G�GJI:KLACMXO

nG:ODA{W_:MP=lKLOQnQACW

n = n0 × n1?:OjGQ?BA

n0, n1 > 1`,J=

n0G:ODA{Ww_:MP=lKLODA{W

n0 = n00 × n01?BOjGQ?:A

n00, n01 > 1 i Wx= n1G:O\A{W�_:MP=�K>O\A{W

n1 = n10×n11?BOjG,?BA

n10, n11 > 1n i ;jWX�QWxE,sdACWX=�Y�;jKLANGgRXAeQ;[W�RT;FANGQsNOjGgRXMT;VM�RPOB?BO[W_:MP=lKLO[WC`��G)ACWXA^KLOjKLACG[RPOze,;Vf:MPANKLO[WZWP;jsN;[?BO

n = p1 × ... × pk?BO[GQ?BALsN;[?:;

piACWZ_:MP=lKLOQ`��WXALMXA{WxEQU RT;j?BOzUlU@;VKD;VKLOgW

E:GQ;T�d�j²N� ¦S�}¯��{�j²N¯65S�8¥N� © �}¯±¤D¦�ª}` �RXMT;�_,ACMPWXOjG,;�ANKL_6AN«C;jGQ?BO)sdOjG¡ACU�KL=@WxKLOn_:E:AC?:AzWP;jsN;jMkOjRXMT;Hp4;jsdRXOjMP=�«C;jsN=lt[G

n = q1 × ... × qj?:=�WxRX=�GgRP; ?:A�U@;

_:MP=lKLANMT;:`J��OjMAx��ANKL_:U�O72 = 6 × 12 = (3 × 2) × (2 × 6) = (3 × 2) × (2 × (3 × 2))i72 = 18× 4 = (2 × 9) × (2 × 2) = (2 × (3 × 3)) × (2 × 2).wGQ;>p4;[s}RXO[MX=�«C;[sd=�tjGmANGH_:MP=lKLOgW?BA �V�>A{W

3 × 2 × 2 × 3 × 2n i OVRPMP;>A{W 2 × 3 × 3 × 2 × 2

` � OVRP;jKLO[W�_:MP=lKLACMXODagE:AsN;[?:;>E:GQ;LRP=lACG:A � _:MP=lKLO[WCn:_6ANMPODKD�jWCh,J=<U�O[Ww_:MX=�KLO[WWXA\_,O[G:ANG�ACG�OjMT?BANGmG:O0?BA{sdMPACsN=lACG[RPAjn:U@;jW�?BO[W�p4;[s}RPOjMP=l«{;jsd=�OjGQACW�WxO[G=�?:�NGgRX=@sN;[WNh

2× 2× 2× 3× 3ngOkKD�jW�sdO[KL_Q;jsdRP;VKLACG[RPAjn

23 × 32`J��UbWxACG[RP=�?:O^?BACUQRPANOjMPANKD;opqE:GQ?:;jK>ACGgRP;VU6ACW�agE:A�eQ; i E:GQ;:n

i ANGHA{WxA\WXANGgRX=@?BODWxt[UlOLE:G,;:nJp4;jsdRXO[MX=�«C;[sd=�tjG�ACGz_QMX=�KLO[W?BA\E:GHGJI:KLANMPO n > 1`�U:O[MXKD;jUlKLANGgRPAjh

B�+�CADE+�-F. �B` P�0 G �d�j²N� ¦V�}¯��{�j²N¯65S� ¥N� © �}¯±¤D¦�ªKJ 3�� [¨j�H�,£B¤L¥N�X¦�¤L�&j¦S�0®N�Q¥��m� ¯4¥N�6¥>�B�� d�[²d� ¦S�}¯��{�[²d¯65V� ¥N� © �}¯±¤L¦Sª 3 ��ªd��d�j²N� ¦V�}¯��{�j²N¯65S��¥}ªD£B�,¯4²P� ¥d� ¥d�rªN¥N�Q�u¯4¨j¦³®d�Q¥DªT¯p1, ..., pk

q1, ..., qj

ªN¦S� © �}¯±¤D¦�ª 2 ²T¦S� pi ≤ pi+1

qi ≤ qi+1© �S�X�²T�j¨j�

i 2 p1 × ... × pk = q1 × ... × qj 2¥d�,� ¦V�6²T¥}ª

k = j

pi = qi© �S�X�z²P�[¨j� i 3�<;)?BANKLO[WxRXMT;jsN=lt[G ?BAHACWxRXO´?BAC_,ACGQ?BA�?BANU��¥N¤L�¡¨[¥ ��Q²N� ¯4¨j¥dª¡y4agE:AHANMT; EQG�U�ANKD;³ANG ANUsdO[G[RPAdcJRXO)ANG agE:AHUlO)EQWXt

��EQsNUl=@?BACWCn�_,ACMXOmagE:AkA{W�E:G³=lKL_6OjMXRP;jG[RPA�� ¥T¦S�X¥N¤L�m?BACU�RXANO[MX�@;0?BA>GJI:KLANMPO[WT|}`,��U��2ACKL;z?BAk��EQsdU�=�?:ACWZACWZsdO[GQWxA{sdE:ACGQsd=@;�?:A

��8��E��1�?� �?�E��?�� E� ��M��#� �?��

OVRPMXO>RPANOjMPANKD;>agE:AFRX=�ANG:A\KÊQsTeQ;[W;V_:U�=�sC;jsN=lO[G:ACWC`J��W�RPAFRXACOjMPANKD; i U�;L?BACKLO[WxRXMT;jsd=�tjGm?BANU��2ANKD;LACGzfQ;[WxA\?BA\ACUlU�ODYjACMXACKLO[W?BA{Wx_:EQ�CWC`:�2ODagE:A\G:ACsNACWX= RT;VKLO[W�;[a[EQ�bACWE:G�sNOjMPOjU@;VMP=lOk?:ANU<�<ANKD;:n:agE:Ao;jGgEQGQsd=@;VKLO[W i ?:ANKLO[WxRXMT;VKLO[W�;je:OjMT;:`

* +�-F. �B` R�0#G ��Q²d� ¯4¨j¥dª J 3 �6¯ p ¥dª © �}¯±¤L¦ �¨S¯±°{¯4¨j¥D�0�B� © �P¦C¨V�Q²d� ¦ ab 2 ¥N�Q� ¦S�6²T¥dª\¦ p¨S¯±°{¯4¨j¥D�

a¦

p¨S¯±°{¯4¨j¥D�

b 3� C D�C &. DE:>C �B` ¸ � 0 �,¯

p¥dª © �}¯±¤L¦ �¨S¯±°{¯4¨j¥L� q1 × ... × qn

¨j¦V�6¨j¥>²T�j¨[�qi¥dª © �}¯±¤L¦ 2 ¥N�Q� ¦S�6²T¥dª p = qi

© �V�X�z�V� j£B� i.4 ¥d¤D¦�ª}�u�X�j²N¯65S� 3 ��OjM�=�GQ?BE,sNsd=�tjG0WXOjfQMXA n `gB= n = 1

U@;Fe:=l_6tVRPACWX=�W�?B=@sdAp = q1

GQOoKD�jW i G:OkagE:AC?Q;FGQ;j?:;F_,;VMT;Z?:ANKLO[WxRXMT;VM{`J=n > 1 i p 6= q1

ANGgRXO[GQsdA{Ww_6OjM�ACU��<ANKD;0?BA^��EQsdU�=@?BACWp?B=�YJ=�?BA>;

q2 × ... × qn`,��G[RPOjGQsNACW�Wx=�U�;0;[WxACMPsN=lt[G�WxA>sdE:KL_:U�A

_Q;jMP;n − 1

np = qi

_,;VMT;0;VU�vjI:Gi > 1

n,A{W�?:ACsd=�M{n�WX=�WxALsdE:KL_:U�A>_Q;VMT;n − 1

ACGgRXOjG,sdACW�Op = q1

Op = qi

_Q;jMP;0;jUlv[I:Gi > 1`:��GgRPOjGQsNACWWXAFsNE:KL_:U�A\_Q;VMT;

n.�

��OjGHACWxRXOL_6OB?BANKLO[WWP;jsC;VM�U@;78+�- C *��D�.�=?:>@A; ?BACU�RXACOjMPANKD;z?:A>U@;zp4;jsdRXOjMP=�«C;jsN=lt[G�ANG´_:MP=�K>OgWN`��r;VMT;mWxE)AdcB=@W�RPANGQsN=�;Qn2WxE:_6OjGQv[;ma[EQA>e,; i GJI:KLANMPO[WFWx=�Gp4;jsdRXO[MX=�«C;[sd=�tjG�ANG³_:MP=lKLO[WCn i WxA{; n

ACU�ACUlACKLANGgRXOzK>��G:=�KLOm?BA>ACWXALsdOjGV��E:GgRXO�GQOzYS;jsN�lO,`,��G[RPOjGQsNACWn = ab

?BO[GQ?BAL;VKkf6O[W?BA

a i bWXOjG)KD; i O[MXA{WZagE:A ¸ n i _6OjMFRT;VGgRXOHKLANG:O[MXA{W\agE:A n

`2��OjM\U@;mK>=�G:=�KD;VU�=�?:;[?´?BAnn<sN;[?:;mE:G:O�?BA

a i bRP=lACG:ANG

p4;jsdRXO[MX=�«C;[sd=�OjG:A{W�ANGo_:MP=�K>OgWN`C��ANMPOANGgRXO[GQsdA{W<WXE\_:MXOB?BE,s}RXOwpqOjMPKL;wE:GQ;�p4;[s}RXO[MX=�«C;[sd=�tjGFANG\_QMX=�KLO[WCnCagE:A�A{W2E:GQ;�sdO[G[RPMP;[?B=@sNsd=�tjG_6OjMTa[EQA

nA{W�E:GzGJI:KLACMXOLWx=�G0p4;[s}RPOjMP=l«{;jsd=�tjG�ANGm_:MX=�KLO[WC`g��;kIQG:=�sC;^sC;VEQWP;^_,OgWx=�f:U�AZ?BAFA{W�RT;^sNOjGgRXMT;j?B=@sNsN=lt[GDACW�U@;^;[WxEQGQsd=�tjG

?BA\U@;>AdcB=�WxRXACGQsd=@;L?BAFGJI:KLANMPO[WWX=lGzp4;jsdRXOjMP=�«C;jsN=lt[G�ACGm_QMX=�KLO[WC`�r;VMT;zU@;mE:G:=@sd=@?:;j?2n2WxA{;

p1 × ... × pk = q1 × ... × qjEQG´sNOjGgRXMT;VAx��ANKL_:U�OmsNOjG

k ≤ j i kU�OHK>ACG:OjM\_6O[WX=�f:UlA[` J=

p1 < q1ANGgRXO[GQsdA{W�_6OjMZANU�sdO[MXO[U�;jMX=�OD;VU��2ANKD;z?BAk��E,sdU�=�?BA{W

p1A{W�EQG:Oz?BA>U�O[W

qin6U�OzsdE,;VU�ACW�=lKL_6O[WX=lf:U�A>_,O[MPagE:A

q1 ≤ qi_Q;jMP;>sN;[?:;in i0i ;âgE:A p1 < q1

A{WxOL?B=@sdA\agE:Ap1 < qi

_Q;jMP;>sN;[?:;i`B��U2KL=�WXKLOL;jMXv[E:KLANGgRXO^KkE:A{W�RPMP;>agE:AFG:OL_:E:A{?BAoWxACM

q1 < p1`J��GgRXOjG,sdACW

p1 = q1i sdO[GQsdU�E:=�KLO[W�agE:A p2 × ...× pk = q2 × ...× qj

`J��OjM�U�;kK>=�G:=�KD;VU�=�?:;[?�?BAknp2 × ...× pk =

q2× ...×qjG:OosdO[GQW�RP= RPE i A�E:GDsNOjGgRXMT;VAx��ANKL_:U�OQ`S��W�?:ACsd=�M{n j = k

n i pi = qi_Q;VMT;�RXOB?BO

in1 < i ≤ k

` UQ;jU RT;\;VU�Ul�QWXtjU�OF?BACsN=lMagE:A

p1 = q1n:_6ANMPO0;>ACWXO i ;>UlO>RPANG:ACK>OgWNn,;jWX�2a[EQA\ANU<WXE:_:E:A{W�RPO0sdO[G[RPMP;jA��ACK>_QUlO>G:ODU�ODACWCnBE:GQ;DsdO[GgRXMT;j?B=@sNsN=lt[G�=�K>_QUl=@sN;[?:;_6OjMU@;>;[WxEQGQsd=�tjGH?BAosdO[GgRXMT;VA��ACKL_:UlOgWN`

�

¢�E:AC?Q;0_Q;jMP;D?BACKLO[WxRXMT;VM�ACU��<ANKD;0?BA>��EQsNUl=@?BA{WN`6��WZE:GQ;0sNOjGQWXACsNE:ANG,sd=@;0?BAkEQG�RXACOjMPANKD;DsdO[G:OBsd=@?BO0fQ;V��O�ACU�G:O[Kkf:MPA?BA ��D¯4¨[¥d�,� ¯4¨[�j¨m¨j¥ I ¥ �{¦S�B�un:agE:A\ANGHMPAC;VU�=@?:;j?z?BANf6A\UlU�ANYS;jMANUbG:O[Kkf:MPA\?BA\��EQsNUl=@?BACWw_,O[MMP;j«NOjGQACW�agE:AFY�;jKLO[W;>YjACMC`

B�+�C DE+�-/. �B` ¸j¸�0 G ¨j¥N�Q�u¯4¨j�[¨´¨j¥ I ¥ �{¦S�B� J 3 �,¯ a, bªd¦V� �,£B¤L¥N�X¦�ªD��S�u�B�X�S�l¥dªD�6¦)�S¤SRT¦�ª��)¥d�,� ¦V�6²T¥}ª�¥P¶[¯@ªT� ¥N� �Q£B¤D¥N�X¦�ª

¥N�Q� ¥d�X¦Sªs, t� �S��¥}ªk®N�Q¥

sa + tb = mcd (a, b) 3��U8��OjGQWC`,µ�AC«NOjE:RwUlO�?:ANKLO[WxRXMPtD_Q;VMT;>_,O[Ul=�G:O[K>=�O[WCn i ANGgRPOjGQsNACWKLANMPACsNA\MXA{sd=�f:=�MwANU�sdMP�C?B=lRXO,nQ_6ANMPO0;jagE:�<GQO[W=�GgRXANMPACWP;WXtjU�Ok_Q;jMP;^GJI:KLANMPO[WCn i A{WxO>ACW�E:G,;^sNOjGQWXACsNE:ANGQsN=�;^=�G:KLAC?B=@;SRT;^?BACU� �� g¦S�}¯±�u¤L¦ ��¶j� ¥N�6¨S¯4¨[¦�¨[¥ ��Q²N� ¯4¨j¥dªoagE:A\?BANKÊ:ACWxRXMT;oU@;AdcB=@W�RPANGQsN=�;L?BA

s, t_,O[M_:MPOJ?:EQsd=�MXU�O[WCn i U�=�WxRXO,`

� &�� &�� CADE:��- C��E+�; $A:�$AC $ +�� = &1:�$ +�* 0st(a, b) =9C1

(b = 0) (1, 0)9C1!"�39 � 7

(c, r) = cr(a, b) � (s, t) = st(b, r)(t, s − ct)

�<;)YS;VU�=�?:AN«�?BA�ACWxRXA�;jUlv[OjMP= RPK>O)WxA�?BACKkE:A{W�RPMP;)_6OjML=�GQ?BEQsCsd=�tjG WxO[f:MPAzACUGgIQK>ACMXO´?BA�U�U�;jKD;j?:;[W^MPACsNE:MTWx=�YS;jW>?BAst`

��MP=lKLACMXO,n,A{WxOzACW�]QG:=lRXOm_,O[MPagE:A>sN;j?Q;0GJE:ANYS;�UlU@;VKD;j?Q;0MPACsdEQMPWX=lYS;LRP=lACG:A>ANUrWxACvjE:G,?BO�;VMPvjEQK>ACGgRXO0KLANGQOjMFa[EQAÂCU�;jGgRXANMP=�OjM_6OjMTa[EQA

resto(a, b) < b`:J=�G:O>eQ; i G:=�G:v[E:GQ;ÛlU@;VKD;j?Q;oMPACsdEQMPWX=lYS;kANGgRXO[GQsdA{W b = 0

n i ANG�A{WxAFsN;[WxO (1, 0)ACW�E:GQ;^MPAN_:E:A{W�RT;

sdO[MXMPACsdRP;\_6OjMTagE:A1× a + 0× 0 = a = mcd(a, 0)

`BB=6e,; i EQGQ;oU�U�;jKD;j?:;kMXA{sdE:MTWx=�YS; i ANU6RXOjRP;jU6?BAZU�U�;jKL;[?:;jW�MXA{sdE:MTWX=lYS;jW�ACWn + 1nJANGgRPOjGQsNACW�ACWP;o_:MP=lKLACMP;kUlU@;VKD;j?Q;oMPACsdEQMPWX=lYS;oRX=�ANG:A

n`:J=�ACWP;kI:UlRX=�KD;^sNOjGgRXA{W�RT;

(s, t)ACG[RPOjGQsNACW�ANUb;jG[RPANMP=lO[M�sNOjGgRXA{W�RT;

(t, s− ct) i U@;LsdE:A{W�RP=lt[G�ACWWX= ta + (s − ct)b = mcd(a, b)`:�we:O[MP;Qn

a = bc + rn:OLWxA{;

r = a− bcn i _6OjM�RP;jGgRXOQnBWX=�RXOB?:;=�GgY[OBsN;jsN=lt[Gm?:ANU�;jUlv[OjMP= RPKLOL?Q;DMXA{WxEQU RT;j?BODsdO[MXMPACsdRXODsdE,;VGQ?BODe,; i n

UlU@;VKD;[?:;jW�MPACsNE:MPWX=�Y�;[WNnQACWsb + tr = mcd(b, r)

n6O�WXAC;sb+ t(a− bc) = mcd(a, b)

y4MXA{sdE:ACMP?BAwagE:Amcd(a, b) = mcd(b, r)

|dngO^WxA{;ta+(s− tc)b = mcd(a, b)

`J��WX�Qe:ACK>OgW�YJ=@W�RPOagE:AFU�;DsNOjMPMXA{s}RP= RPEQ?�?BACU<;VU�vjO[MX=lRXKLOLsdEQ;jGQ?BOLeQ; i n

MXA{sdE:MTWX=lO[G:ACW�=�K>_QUl=@sN;LWXEHsNOjMPMXA{s}RP= RPEQ?0_Q;jMP;n + 1

MPACsdEQMPWX=lO[G:ACWC`��WX�<agE:AC?Q;>?:ANKLO[WxRXMT;j?:;Û@;^=@?BACG[RP=�?Q;j?H?BAoµ�AN«COjEBR{`��W=�GgRXANMPACWP;VGgRPAZGQOVRP;jM�OVRXMT;L?BANKLOgW�RPMP;[sd=�tjGz?:AZU@;>=�?:ANGgRX=@?:;j?H?BAoµ�AC«NOjE:RC` � ;j?BOgW

a, bGQ;VRXE:MT;VU�ACWCnJG:O0;VK^f,OgW

0nQWXAC;

C = {sa + tb : s, tACGgRXANMPO[WCn

sa + tb > 0} `JA{;mANU�ANU�ANKLANGgRPOzKL��G:=lKLOm?BA

CyCG:OmACW�YS;jsd��O�_6OjMTagE:A^sNOjGgRX=�ANGQA

a + b > 0| i WxA{; g = mcd(a, b)

`��OB?BOz?:=lYJ=@WxO[MsdO[KkI:G

d?BA

a, b?B=�YJ=�?BAZ;

m_,O[MPagE:AZsdOjG,Wx=@W�RPAwANG�U�;^WxE:KD;^?BA�?BOgW�KÎ:U RP=l_QUlOgW�?BA

d`J�r;VMT;\WX;jf,ACM�agE:A

m = gnjp4;jU RT;kWP;Vf6ANM

WX=�ACW^?B=�Yg=@WXOjMksNOjKkIQG8?BAa, b`�JE:_6OjG:vg;�a[EQA

mG:O ACW\p4;[s}RPOjMk?BA

a`<��GgRXO[GQsdA{WoANcJ=@WxRXANG

c, rRP;VU�agE:A

a = cm + rsNOjG

0 < r < m`B��GgRXOjG,sdACW�A{W

r = a − cm = a − c(sa + tb)_Q;jMP;>;VU�vjE:GQO[W�ANGgRXACMXOgW

s, tn:ODWxA{;

r = (1 − cs)a − ctbn:?:;jGQ?BO

agE:Ar ∈ C

n6_6ANMPOmA{WwANU�ANU�ANKLACG[RPO0KL�lGQ=lKLO�?BA

C`,��G[RPOjGQsNACW�Wx�

mACWwp4;js}RPOjMZ?BA

a`�B=lKL=�U�;jMXKLANGgRPAjn

mACWwp4;js}RPOjMZ?BA

b`

��GgRXO[GQsdA{WNnB_6OjM�U�;>_:MPOj_:=�AC?Q;j?�agE:A\YJ=�KLO[W?BA{Wx_:EQ�CW?BA\U@;L?BAd]QGQ=�sN=lt[Gm?BAmcdnm = g

`

��

� tVRPACWXAZagE:A\sNEQ;VGQ?:OkANU2;jUlv[OjMP= RPKLOoUlU�ANvg;^;b = 0

_:E:AC?:AFsNOjGgRXA{W�RT;VM(1, n)

_,;VMT;ksdE,;VU@a[EQ=lACM�ACGgRXANMPOn_,O[MPagE:A

1× a +n× 0 = a = mcd(a, 0)

n i a[EQAFsC;j?:;Ê:G:OD?BAFA{W�RPO[W�?:;^MXA{Wx_:EQACWxRP;>?B=�WxRX=�GgRP;L;VU�_:MP=�K>ACM�G:=lY[ANUu`B��UbA��ACMPsN=�sN=lO ¸C¹ ;S],MXKD;>a[EQA;jWX�2WxA\sNOjGQWX=lv[E:ANGzRXOB?BOgW�UlOgWs, tagE:A\eQ; i `�we:O[MP;H_Q;jMP;H?BACKLO[WxRXMT;VM\ANU��2ANKD;�?BA��EQsdU�=@?BACWCn�WXE:_6OjG:vg;VKLO[WFagE:A

pG:O�?:=lYJ=@?BA�;

a`<��GgRXO[GQsdA{WNn�Wx=�ANGQ?:O

p_:MX=�KLOQn

ACWmcd(p, a) = 1

n i ACGgRXOjG,sdACW�_6OjMZU�;z=�?:ANGgRX=@?:;j?³?BA^µ�AC«NOjE:RZANcJ=@WxRXANG s, tRP;jUlA{W�agE:A

sp + ta = 1`��E:UlRX=�_:Ul=@sN;jGQ?BOmACWxRP;

ACsNEQ;jsN=lt[G�_,O[MbWP;VU�A

spb + tab = bn imi ;0agE:A ab i _6OjM�RP;VGgRPO tab

ACWwKÎ:UlRX=�_:UlOm?BApnbA{WwU@;�WXE:KD;0?BA>?BO[W�KkI:UlRX=�_:U�O[W

?BAp i ANGgRPOjGQsNACW b A{WKkIQU RP=l_:U�OD?BA p

`,��WX�bYjANKLOgWa[EQAFWX=pG:OD?B=�Yg=@?BAo;

aRP=lACG:AoagE:A\?B=�Yg=@?B=�M�;

b`

��OjGmACWxRXOQnQU�;L?BACK>OgW�RPMP;[sd=�tjGm?BANU�RPANOjMPANKD;^pqE:GQ?Q;VKLANGgRP;jUbACWxRP�LsdO[KL_:UlANRP;Q`� ��

�� 1 � 1 3,9 �2+ �¸ ` y ¸ |o��;jU�sNE:UlA

st(a, b)_,;VMT;

(a, b) = (12, 16)n(4, 10)

n(9, 24)

n(14, 65)

n(28, 130)

n(108, 188)

n(3214, 303) iY[ANMP= ]6a[EQAFsC;j?:;>MPACWXE:UlRP;j?:OQ` � OLACWwG:ACsNACWP;VMP=lO^K>OgW�RPMP;jM�UlOgW?BAdRT;VU�UlA{W?BANU<sNOBsd=�ANGgRXA i MPACWxRXO,`�B` yu�j|��N�r;jMP;\sNEQ�VU�ACW�Y�;jUlO[MXA{W�?BA

kWxA�sNE:KL_:U�A�agE:Ajn[_Q;VMT;FRPOJ?:Oo_Q;VM�?BAwGQ;VRXE:MT;VU�ACW

a, bna) mcd(a, b) = mcd(a, b+

ka) � b) mcd(a, b) = mcd(a, a + kb)`

¹ ` yu�j|w��WXsNMX=�fQ;LACG�G:OVRT;jsN=lt[GH;jUlv[OjMP� RPKL=�sC;LE:G�_:MPOJsNAC?B=�KL=lACGgRXO0MPACsNE:MPWX=�YjODagE:AoMPACsd=�f6AksdO[K>Oz;VMPvjE:KLANGgRPO[W�U�O[WwG,;SRx�EQMP;jUlA{W

a i pn i ANGQsNE:ANGgRPMP;ÂNU2KD�VcJ=�KLOLGQ;SRPE:MP;jU n i E:GzG,;SRXEQMP;jU k RT;VU�ACWagE:A a = k · pn

`� ` yu�j|w��WXsNMX=�fQ;�E:G�_:MPOBsdAC?:=lKL=�ANGgRXO0_,;VMT;DANGQsNOjGgRXMT;VM�U�;0?:ACWPsdOjKL_6O[WX=�sN=lt[G�ANG�_QMX=�KLO[W�?:A^sdE,;VU@a[EQ=lACM�GQ;VRXE:MT;VU

a`6��U

_QMXOBsdA{?B=�K>=�ANGgRPOL?:ANf6AosdOjGgRPACWxRP;VM�E:G,;Û�=@W�RT; {(p1, n1) , . . . , (pk, nk)} RP;VU�a[EQAa = pn1

1 · · · pnk

k

sNOjGp1, . . . , pk

_QMX=�KLO[W i p1 < · · · < pk`

� ` y ¹ |�S¢�E:�F_Q;[WX;LsdO[Gst(2a, 2b)

�o�st(3a, 3b)

�o�st(ka, kb)

�o�C��OjM�agE:��:` y ¸ |�C�rANMT?:;[?BANMT;kO>p4;jU�WP;[�

a2?B=�YJ=�?:Ao;

b2Wx= i WxO[UlOLWx= a ?B=�Yg=@?BAo; b

`�J` y ¸ | � ANKÊ:ACWxRXMPA\a[EQA

mcd(ka, b) = mcd(a, b)Wx=

k, bWXOjGHsdO[_:MX=�KLO[WC`

P ` yu�j| � ANKLO[WxRXMT;VM{h:J=a i b

G:O0WXOjGH;VK^f,OgW��:n abmcd(a,b)

ACW�ACU2KL�lG:=�KLO0sdO[KkI:GmKkI:UlRX=�_:U�O0?BAa i b`

R ` yu�j|��C�rANMT?:;j?:ANMT;oO^p4;VU@WX;g�ZB=rACW�MT;jsd=�OjG,;VUungsC;j?:;

aiA{W�ACGgRXANMPOQn i rn + an−1r

n−1 + an−2rn−2 + · · ·+ a0 = 0ACGgRXOjG,sdACW

rACWACGgRXANMPOQ`

¸ �:` y ¸ | � ANKLO[WxRXMT;VM{h:J=a i b

WXOjGHsdO[_:MX=�KLO[W i abACWE:GHsdE,;j?BMT;j?BO>ANGgRPOjGQsNACW

a i bWXOjGHsdEQ;[?BMT;j?BOgWN`

¸j¸ ` y ¹ |Z��;VMT;js}RPANMP=�sNAZRXOB?BO[W�UlOgWRPMX=@�VG:v[E:U�O[W�MPACs}RPO[W�sdE i O[WU@;j?BOgWwWXOjG�GJI:KLANMPO[WG,;SRXEQMP;jUlA{WN`,BE:vjACMXACGQsd=@;:hQ?:ANKkEQACWxRXMPAagE:AoWX=x2 + y2 = z2 i x, y, z

G:OLRX=�ANGQANGmp4;jsdRXOjMwsNOjKÎ:GHANGgRXMPAFUlOgW�RXMPACWCnBANGgRXO[GQsdA{Wx, yG:OD_:EQAC?BACGHWXANM�;VKkf6O[W

=�KL_Q;VMPACWFy ?BANKÊ:ACWxRXMPA i ;V_:U�=�agE:AZANUbU�ANKD;Qh resto(a2, 4)ACWWX=lACKL_:MXAF�Ô ¸ ` |��GgRXO[GQsdA{W�WxA{;

y = 2w i sdO[GQWX=�?BACMXAagE:Aw2 = z2−x2

4

`:�w_QUl=@agE:A\ANU2Ax��ANMTsd=@sd=�O ¸ �>_Q;VMT;^sNOjGQsNUlEQ=lMagE:A\_Q;jMP;>;VU�vjE:GQO[W�GQ;SRPE:MT;VU�ACWm, nnQsdO[_:MP=lKLO[W i GQO;jKkf6O[W�=�KL_Q;VMPACWCnBACW

z = m2 + n2nx = m2 − n2

n i y = 2mn`

¸ �B` yu�j|DJA{;VGa = pn1

1 · · · pnk

ki b = pm1

1 · · · pmk

k

?BO[WFACGgRXANMPO[WF_6O[WX=lRX=�YjO[WDy ;VU�vjE:GQO[Wmii QSO ni

_,OB?BMP��;jG¡WxACM0|}`

��E:ACWxRXMPA\a[EQAjh�[| y �j|

a|b WX= i WXtjU�OLWX= ∀i ∈ {1, . . . , k} , ni ≤ mi`

RC| y �j|mcd (a, b) = p

mın(n1,m1)1 · · · pmın(nk ,mk)

k

`¸C¹ ` y ¹ | � ANKÊ:ACWxRXMPAwagE:AwsdE,;VU@a[EQ=lACM

s, tagE:A�sdE:KL_:U�A

sa+ tb = mcd(a, b)WXAwsdO[GQWx=�vjEQA_,O[M�ANUQ;jUlv[OjMP= RPKLOZANcJRXANG,?B=�?:O

?:A\��EQsdU�=@?BACWKLOB?B=l],sN;[?BOL_Q;VMT;>sdOjGgRPACWxRP;VM(1, n)

ACGzACU2I:UlRX=�K>ODGQ=lY[ANUu`� ��

�<+ - � 1��C5��{1 � 7��3 � � � 7 ��2+ *,+ � � � � 3�+ J=�E:GQO0RXMT;SRT;0?BALsN;jU�sNE:U�;jM

resto((n − 1)!, n)_Q;VMT;

nsTe:=@sdOzG:O�e,; i _:MPOjfQUlACKL;Q`Q��ANMPO�_,;VMT; n =

10000G:=A� ;V_:U�A�_QE:AC?BA\sNOjGgRXACG:ANM�E:GmGgIQK>ACMXO^RP;jG�v[MP;jGQ?BAFsdOjKLO R;R;R0R $�`[��ACMXO

resto(ab, n) = resto(a× resto(b, n), n)yqY[ANMPANKLO[W�ANG,WxACvjE:=@?:;>_,O[MwagE:�{|d`:��GgRXO[GQsdA{WNnQsNEQ;VG,?BOLUlU�ANvg;VKLO[W�;7! = 5040

nJsN;jU�sNE:U�;jKLO[Wresto(8 × 5040, 10000) = 320i WP;Vf6ANKLO[WDagE:A resto(8! × 9 × 10 × ... × 999, 10000) = resto(320 × 9 × ... × 999, 10000)

n i agE:A�ACUI:U RP=lKLO¡A{Wresto(resto(320 × 9, 10000) × 10 × ..,9999, 10000) = resto(2880 × 10 × ..,9999, 10000)

`$��;je:OjMT;³sC;VU@sdE:U@;VKLOgWresto(10 × 2880, 10000) = 8800 i ACUw_:MPOjf:U�ANKD;)WxAHMPAC?BE,sdA�; resto(8800 × 11 × ... × 9999, 10000)

n<UlO¡sdE,;VUwsdO[Gresto(8800× 11, 10000) = 6800

WxA\MPAC?:EQsdA\;resto(6800× 12× ... × 999, 10000)

`J��U@;VMPOQngRPOJ?Q;�Yg�@;kp4;VUlRP;VGHE:G,;jW R;R0P;P= RPANMT;jsN=lO[G:ACWCn�UlO�agE:ALG:O�agE:=�WX=��NMT;VKLO[W�eQ;jsNANMo;zKD;VG:O,nb_,ACMXO ��;j_:UlAL_:EQAC?BALeQ;[sdANM\A{WX;[WZ= RPANMT;jsd=�OjGQACWZANG)E:GQ;zpqMT;jsNsN=lt[G)?:AWxACvjE:G,?BO^_6OjMTagE:A\Wxt[UlOL=�GJYjOjU�EQsNMP;kGgIQK>ACMXOgW?BA � ?B��vj=lRXO[W�OLKLANGQO[WC`B��U2_:MPOjv[MP;jKL;kANG ��;j_:UlAFA{W

prod := 1 : for m from 2 to n − 1 do prod := (m ∗ prod)KLOJ?

n od

agE:Ao?BA��x;LACU2MPACWXE:U RT;j?BO>ANGprod`,��MPOjf,;VGQ?BO>ACWxRXO,n:YS;D;>YjANMwagE:A\_,;VMT;

n = 10000e,;jWxRP; ¸ �[�j� >?:;D�Qn i _Q;jMP; n =

¸ �j�[��?:; ¸ �[�j� Q`g�<OD?BA��OD_Q;jMP;^_6OjGQ?BACMP;jM�ACWxRXA\e:A{sTe:OQ` � ACWX_:E:�{W�Y�;jKLO[W;>YjACMwa[EQ�FWX=�vjG:=l],sN;Q`wGLA��ACK>_QUlOFANGLagE:AWxO[UlOZ=�G[RPANMPACWP;�ACUBMXA{W�RPO�A{W�sdEQ;jGQ?BOZRXMT;VfQ;V�x;VKLO[W�sdO[G^�jG:vjE:U�O[WC`VJ=Bvj=�MPOZ;ZU@;�?BACMXA{sTeQ;F�V�j��vjMT;j?:O[WCn iUlEQANvjO ¸ �j��KD�[WNn i UlE:ACvjO �S�ZKD�jWCnSe:A�e:ACsTeQOZEQG>v[=lMPOF?BA resto(390, 360) = 30

v[MP;[?BO[WC` � A{sd=�K>OgWragE:A�� ¹0R ��ACW�ACagE:=�YS;VU�ANGgRXA; ¹ �>KLtJ?:E:UlO ¹ j��}`J��WE:G,;>WX=lRXEQ;[sd=�tjGmANG�agE:A ¹;R �ÂCWw=lGQ?:=�WxRX=�G:vjEQ=lf:U�Ao?BA ¹ �Q`

�)��#� �?�� E�� xº��G>ANU:sdEQANGgRP;!(g=�U�tjKLAdRPMXOgW<?BA�E:GL;VEBRPOQn ¸C¹ �j�j�[�j��A{W�=�GQ?B=@WxRX=�G:vjE:=�f:U�A�?:A ¹ �j�[�j�j�Q`d��G>E:GQ;ZsdO[K>_QEBRP;[?BOjMT;�sNOjG^;jMX=lRXKL�NRX=@sN;

?BA ¸ ³f:=lRPWCn� �!� � �216 + 10

ACW>=lG,?B=�WxRX=�G:v[E:=lfQUlAH?BA ¸ �:`��ANMPO³WX�WX;jf,ACKLO[WLagE:Az_6OjMLKD�jW>;j?:=�sN=lO[G:ACWCnrWxEQWxRXMT;jsCsd=�OjG:A{W i_:MPOB?BEQs}RPO[WagE:AFeQ;[sdANKLOgWNnBACUbGgIQK>ACMXODagE:AFMPACWXE:U RT;>ACWANcB;[s}RT;VKLANGgRXAZACagE:=�YS;VU�ANGgRXAFKLOB?BE:U�O216 ;VUbMPACWXE:U RT;j?BODsNOjKL_:U�AdRXO,`�<;¡_,;VU@;Vf:MT; ¤85{¨S�B��¦´pqE:A³EQWX;[?:;´_6OjM�_QMX=�KLANMT;ÝjAN« _,O[M �F;VE,WXWCn�ANG 4 ¯@ªN®N�B¯@ªT¯4²N¯4¦S��¥}ª}`��=�ANGQA ?BA�E:G YjANMPf6O¡U�;VRX=�G:O

agE:AkWX=lv[G:=l],sN;o�xKLAC?B=�M��}n:_6ANMPO �F;VEQWPWwWXANv[E:��;D;L��EQsdU�=�?:ACWZa[EQ=lACG2n,sNOjKLO0?B= ��=lKLO[WCn,?:=�WxpqMP;j«C;jfQ;>U�;>RPANOjMP�@;D?BAoGJI:KLANMPO[WwsNOjKLOvjACOjKLAdRPMX�@;:n i ?BA{sd�@;âgE:AZE:GHWxACvjKLANGgRXOÂNMT;oE:G,;�¤D¥T¨S¯4¨j��?BAZOVRXMPO^WX=�ANUbOjRXMPOkANMT;oYS;jMX=@;jW�sdO[_:=�;[W�?:ANUb_:MP=lKLACMXO,n[A{W�?:ACsd=�M{nJWXEU�OjG:v[= RPEQ?�ACMP;>KÎ:U RP=l_QUlODANcB;[s}RPO>?BACU2_:MP=lKLACMXO,`�<;\KLOB?BACU�;[sd=�tjGLKD;SRXACKD�SRX=@sN;\?:AwACWP;jW�WX=lRXEQ;[sd=�OjG:A{W�ANG0agE:A�WXtjU�O\_,ACMPsN=lfQ=lKLO[W�ANUQMPACWxRXO\KLtB?BE:U�O

nACWrEQG:Ok?BAwU�O[WrRXANKD;[W

sdU@;�YjA{W�?BA\U�;>KD;SRPANKD�SRP=�sC;L?B=�WPsdMPAdRT;:`�2+��b+ - � 1 �{5��C1 � 7 583 � � � 7 �� "�7 � � � 7 � 1+ " � � � 3 "�� " � 1472+

7 +�91:<; :>=�:>@A; �:` ¸ � 0 JAD?B=@sdAD?BAD?BOgWFGgIQK>ACMXOgW�ACGgRXANMPO[Wa, b i E:G)GgIQK>ACMXOmGQ;VRXE:MT;VU n > 1

a[EQAa ≡n b

WX=a − b

ACWFEQGKkIQU RP=l_:U�O0?BA

n`

a ≡n bWxAFU�ANAw�

aACWwsNOjG:v[MXEQANGgRXAFsdO[G

bKLtJ?:E:UlO

n�}`

��WXO0ACWA{agE:=lYS;jUlACG[RPAo;D?BACsN=lM�a[EQAresto(a, n) = resto(b, n)

Wx=bRPANG:ACKLO[WwsNE:=�?Q;j?BO0?BA^?BAd]QGQ=lMresto(a, n)

sNOjKLOLANUr?BACUb_Q;VM�I:G:=@sdOL?BAFANGgRPANMPO[W

(c, r)sNOjG

cACG[RPANMPO i 0 ≤ r < |b| RT;VU2agE:A a = cb + r

n:agE:AFG:OÂ{W�RT�L?BA\;jsdEQANMT?BO^sNOjGz;jUlv[E:GQ;[WsdO[KL_:EBRP;[?BOjMT;jW�sdEQ;jGQ?BO

a, bG:O^WXOjGz;VKkf6O[W�GQ;VRXE:MT;VU�ACWCn[;[Wx��agE:A�KLA��O[Ma[EQA�;j?:Oj_BRPANKLO[W�U�;>?BAd],G:=�sN=lt[G�?:;[?:;:`J�<;kMXACU�;[sd=�tjG

≡nA{WE:GQ;m�X¥N�l�[²d¯65V�´¨[¥^¥T®N�B¯±°��V�l¥d��²d¯4�Vn6sNOjGQsNAN_BRPODa[EQAZYS;jK>OgW;>ANGQsNOjGgRXMT;VM�KkE,sTeQ;jW�YjA{sdA{WNn i a[EQAF?:Ad]QG:=�KLO[W;je:OjMT;:h

7 +�91:<; :>=�:>@A; �B` ¸C¹�0 wGQ;H�X¥N�l�[²d¯65V�´¨[¥^¥T®N�B¯±°��V�l¥d��²d¯4��ACW�EQGQ;>MXACU�;[sd=�tjGmf:=lG,;VMP=�;>WxO[f:MPAZEQGmsNOjGS��E:GgRPOCRT;VU2agE:A

¸ `�y , A �QAdcB=�YS;[|��r;VMT;kRPOB?BOa ∈ C

na ≡ a

`�:`�y J=�KL�dRPMX=@sN;g|��r;jMP;^RXOB?BO

a, b ∈ Cna ≡ b

=�K>_QUl=@sN;LagE:Ab ≡ a

`¹ `�y4��MP;jGQWX= RP=lYS;[|��r;VMT;kRXOB?BO

a, b, c ∈ Cna ≡ b i b ≡ c

=�KL_:Ul=@sN;DagE:Aa ≡ c

`�<;H=@?BAC;�?:A0E:GQ;HMPANU@;jsN=lt[G)?BA�ACagE:=�Y�;jUlACGQsd=@;mACWoagE:Az;VE:GQagE:ADU�O[WoANU�ANKLANGgRXOgW\?BA

CWXOjG)ANG¡MPAC;VU�=@?:;j?)?B=@W�RP=lGgRXOgWNn<_,O[M

;VU�vjEQGQ;MT;V«Nt[GFG:Ow_6OJ?:ANKLO[W�?B=@W�RP=lG:v[E:=�M<ANGgRPMXA�;VU�vjE:G:OgW � G:O[W<_Q;VMPACsNANG\=�vjEQ;jUlA{WN`C��O[M<Ax��ANKL_:U�OQn{E:GovjMPE:_6OwANGâ[EQA�eQ; i ;VU�vjE:G:OgW_Q;jMXA{W<?BA�vjACK>ACUlOgW2=@?B�NGgRP=�sNO[WCh{WXEoKD;j?BMPA�U�O[W�_:E:A{?BA�?B=�WxRX=�G:v[E:=lM�_6ANMPO�ANU�U@;wG:O�ACWxRP�:`V ;VU�vjEQG:O[W<f:UlOBagE:A{W�?BA<��E:vjEQAdRXA[hS;VU�vjE:G:OgWsdEQf,OgWNn:A{W�pqACMP;[WNnJ_:=�MT�VKL=�?:ACW i sNOjG:OgW?BA\YS;VMP=lOgW�sdOjU�OjMPACWC`J��GHE:GQ;>pqOjRXOjv[MP;Vpq��;ÂNGHf:U@;VG,sdO i G:ACvjMPO^GQOD_,OB?BACK>OgW?B=@W�RP=lG:v[E:=�ME:G�sNOjG:ODMPOV��O0?BAkE:G�sdO[G:O0;V«CE:U `b��U�;jMXOLA{W�RT�0agE:AkWX=2G:O0_:EQAC?BO�?:=�WxRX=�G:vjEQ=lMwACGgRXMPA

a i bACWw=�vjE,;VU�K>ACGgRXA\YS�VU�=�?:O0?BACsN=lMZa[EQA

G:OL_:E:A{?BOL?B=�WxRX=�G:v[E:=lMACGgRXMPAb i a`J�&WX=bG:OL_:E:A{?BOL?B=�WxRX=�G:v[E:=lMGQ=bANGgRXMPA

a i bG:=bACGgRXMPA

b i cnBA{W�MP;j«NOjG,;Vf:U�A�WXE:_6OjG:ACM�agE:A

G:OLYjO i ;D?:=�WxRX=�G:vjEQ=lMACG[RPMXA a i c`

�2U@;VKD;VKLOgW[a];VU<sNOjGS��EQG[RPOD?BAFRXOB?BOgW�UlOgW�ANU�ANKLANGgRXOgW?BA

CACagE:=�Y�;jUlACGgRXACW;

ah[a] = {b ∈ C : b ≡ a} `B��WwsNU�;jMXOLagE:A

C =⋃

a∈C

[a]`B�?BACKD�jWCn[_Q;jMP;^sN;[?:;

a, b ∈ Cn0WX=

a ≡ bANGgRXO[GQsdA{W�O^f:=lACG

[a] = [b]O

[a]∩ [b]ACW�YS;jsN�lO,`g��Wx��U�O[W

[a]pqOjMPKD;VG

E:GQ; © �S�}�u¯4²d¯65V�)?:A CyqY[ANMAx��ANMTsd=@sd=�O ¸ |}`

J=�eJE:f:=��NMT;VKLOgW\?BAd]QGQ=�?BOa ≡n b

sNEQ;VGQ?:Oresto(a, n) = resto(b, n)

WxACMX�@;zOjfJYJ=lO�agE:ALACW\EQGQ;mMXACU�;[sd=�tjG´?BALACagE:=�Y[�;VU�ANG,sd=@;:n�_,O[MPagE:AHWX=lACK>_QMXAmUlO)ACWLsNEQ;VGQ?:O)WXAm?:Ad]QG:AHACG RP�NMPK>=�G:OgW>?:AzE:G,;³_QMXO[_:=lA{?:;j? sdOjKL_Q;jMxRP=�?Q; ACG[RPMXA

a i b`��r;VMT;

?BACK>OgW�RPMP;jM�agE:A>U�;[WF?BO[WZ?BAd],G:=�sN=lO[G:ACWFWXOjG³ACagE:=�Y�;jUlACGgRXACWCn6G:ACsNACWX= RT;VMP��;jK>OgWwACU�RPANO[MXACKL;z?BA>U�;m?B=�Yg=@WX=lt[G _Q;jMP;�UlOgW�ACG[RPANMPO[W �_6OjMACWXOL_:MPAdpqANMP=�K>OgW�?BANKLO[WxRXMT;VMPU�O>?B=�MPACs}RT;VKLANGgRPAF?:AZU@;D?BAN]QG:=@sd=�tjGH?:;j?:;Qh¸ `

a ≡n a_6OjMTa[EQA

a − a = 0 = 0 × n`

�:`�J=a ≡n b

ANGgRXO[GQsdA{Wa − b = kn

_Q;jMP;>;VU�vjI:GkANGgRXACMXO � ANGgRXO[GQsdA{W b − a = −kn

;jWX�2agE:Ab ≡n a

`¹ `�JE:_6OjG:vg;zagE:A

a ≡n b i b ≡n c`b��GgRXOjG,sdACWF_,;VMT;z;jUlv[I:G

k1, k2A{W

a − b = k1ni b − c = k2n

`��GgRXO[GQsdA{Wa − c = a − b + b − c = k1n + k2n

nBE:GHKkI:UlRX=�_:U�O0?BAn`

�<;mMPANU@;jsd=�tjG ≡nACWZRP;jKkf:=��NG)E:GQ;mMPANU@;jsN=lt[G ?BA ²T¦S� V�}�Q¥N�6²N¯4�HMPACWX_,A{s}RXOH?BA�� i RP;VK^f:=��NG MPACWX_6ACs}RPO�?BA ×,

sNOjGQsNAN_BRPOagE:AFYS;VKLO[W;>MPANANG,sdOjGgRPMP;jM�YS;VMP=@;jW�YjA{sdA{WNh7 +�91:<; :>=�:>@A; �:` ¸ � 0 JA{; ≡ E:G,;kMPANU@;jsN=lt[G�?BAZACagE:=�Y�;jUlACGQsd=@;kWXOjfQMXA�E:GzsNOjGS��E:GgRPO C i WXAC; ◦ E:G,;kO[_,ACMP;[sd=�tjGDf:=�GQ;VMP=@;kWXOjf:MPAC`,JAo?:=�sNAFagE:A ≡ ACWF�B�6�0�X¥N�l�j²N¯65S�¡¨j¥>²T¦S� j�}�Q¥N�6²N¯4�D�X¥dª © ¥T²N� ¦z¨j¥ ◦ Wx=b_,;VMT;kRPOB?BO[W a, b, c, d ∈ C

na ≡ b i c ≡ d

=�KL_:Ul=@sN;DagE:Aa ◦ c ≡ b ◦ d

`�rAC;jK>OgWkagE:A ≡n

A{Wk?BAzsdOjGQvjMPE:ANGQsN=�;HMPACWX_6ACs}RPO ?BA+h(a + c) − (b + d) = (a − b) + (c − d)

n�U@;�WXE:KD;�?BA�?BOgWKkIQU RP=l_:U�O[W?:A

n i _6OjM�RT;VGgRXOLE:GHKÎ:U RP=l_QUlO0?BA n`

�rAC;jK>OgWkagE:A ≡nACWkE:GQ;�MPANU@;jsN=lt[G¡?BA�sdO[G:vjMPE:ACGQsd=@;mMXA{Wx_6ACsdRXO�?BA × h ac − bd = a(c − d + d) − (b − a + a)d =

a(c − d) + (b − a)dn i�i ;âgE:A c − d i b − a

WxO[Gm;jKkf6O[W�KkIQU RP=l_:U�O[W?:Anna(c − d) + (b − a)d

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0, 1, 2, ..., n − 1`<��;�sdU@;jWXAL?:ADACagE:=lYS;jUlACGQsd=@;�?BA

0n[0]n2A{WFRXOB?BOgW\U�O[WFKkIQU RP=l_:U�O[Wk?BA

n`<�<;�sNU�;[WxAD?BADA{agE:=lYS;jUlACGQsd=@;�?BA

in

0 ≤ i < nA{W {cn + i : c

ANGgRXACMXO } `B�rAC;jK>OgW�U�;[W�RP;VfQU�;[W?BAo;j?B=@sd=�tjG i ?BA\KÊ:UlRX=�_:Ul=@sN;[sd=�tjGm_Q;VMT; n = 7 i n = 8h

� � �� ¸ � ¹ � � × � ¸ � ¹ � � � � ¸ � ¹ � � � � �� ¸ ¸ � ¹ � � � ¸ � ¸ � ¹ � � � � ¹ � � �� ¸ � �� ¸ ¹ �¹ ¹ � � � ¸ � ¹ � ¹ �� ¸ �� ¸ � ¹ � � � ¸ � � ¹� � � ¸ � ¹ � � � � ¹ ¸ � � �� ¸ � ¹ � � � � � ¹ � ¸

Z7

� � ¸ � ¹ � � � × � ¸ � ¹ � � �� ¸ � ¹ � � � � �� ¸ ¸ � ¹ � � � � ¸ � ¸ � ¹ � � �� ¹ � � � � ¸ � � � � � � � ¹ ¹ � � � � ¸ � ¹ � ¹ ¸ � � � �� ¸ � ¹ � � � � � � � � �� ¸ � ¹ � � � � � � � ¸ ¹ � � ¸ � ¹ � � � � � � � �� ¸ � ¹ � � � � � � � ¹ � ¸

Z8

�2O[W�G:O[Kkf:MPACWZ7, Z8

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ACagE:=�YS;VU�ANGQsN=�;L?BA ≡nWxO[f:MXA

Zn:ACGH?:OjGQ?BA\?BAN]QG:=�KLO[W

[a] + [b] = [a + b] i [a] × [b] = [a × b]`

��;jMP;Zp4;[sd=�Ul=lRP;jM�U@;osdO[GQWxRXMPEQsNsN=lt[GL?:AwACWxRP;[WrRP;jf:U@;jWrG:tjRXA{WxA�agE:A[njACGDU�;FRT;Vf:U@;o?BA+ngsN;[?:;F]QU@;\ACW�E:GQ;\MPOVRT;jsN=lt[GL?:AwU�;\]QU�;

;VGgRXACMX=�OjM{`��ANGDU@;ZRP;jf:U@;\?BA × nVG:tjRXA{WxAwagE:AwWXAwANKL_:=�AN«{;F_,O[G:=�ANGQ?BO 0ACGDU�;F_:MP=�K>ACMP;FsNOjU�E:KLGQ;\?BAwU@;Z]QU@;

a, i WXA_:MPOJsNAC?BAw;WxEQKL;jMa;FsN;j?Q;ZsNOjU�E:KLGQ;>yqKLtJ?:E:UlO

n_,O[M�WxEQ_:E:ACWxRXOJ|}nS_6OjMTagE:A;jagE:�un�sNOjKLOFANG>UlOgW�ANGgRXACMXOgWNnSWXAsdE:KL_:U�A

a(j+1) = aj+a.

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Z7n6ACW

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_,;VMT;^sC;j?:;a, b?:=�WxRX=�GgRXO[W�?BA

0nB_6OjMTagE:AZ_,OB?BACK>OgW�MXA{WxO[UlY[ANM

ax = 1nby4UlU@;VKD;VKLOgW

a−1;>ACWxRXAx|�_6OjMTagE:A

b = (aa−1)b = a(a−1b)`

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2ACG>U@;�RP;jf:U�;FGQO

;V_Q;jMXA{sdA1 � RP;jK>_6OBsdO 3

O5`��ANMPO

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2sNEQ;VU@agE:=lACM

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5nQsC;j?:;^GJI:KLANMPOD?BA

1, 2, 3, 4, 5, 6, 7;V_Q;jMXA{sdAZE:GQ;LY[AN«j` � A\e:A{sTe:OQnB_Q;jMP;>sN;[?:;

a=�K>_,;VM{n

a2 = 1.B�+�CADE+�-F. �:` ¸ ��0 ��

Zn 2�B�´¥d��¥d¤D¥d�,� ¦

a¨V¯@ª}� ¯±�,� ¦m¨j¥

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aRX=�ANG:A

psdO[KLO�p4;[s}RPOjM{`S��ACMXO ¸

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pn i ;VU�?BACsN=lM�agE:A resto(ma, n) = 1

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cnmaACW�KÎ:U RP=l_QUlO0?BA

p i cn + 1GQO>A{WKkIQU RP=l_:U�OD?BA

p_,O[MPagE:A

cnUlOLA{WN`

��O[M�ANUQOVRXMPOFU�;[?BOQnVWXAC;jGa i n

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sa+ tn = 1`

��GgRXO[GQsdA{Wsa −1

ACWZE:G³KkI:UlRX=�_:U�Om?:Ann�OzWXAC;zagE:A

sa ≡n 1` �rcB=�WxRXA

xn0 ≤ x < n

n6RP;jUra[EQAs ≡n x

_6OjMTagE:A>WP;Vf6ANKLOgWagE:A

Z =n−1⋃

i=o

[i]`,��WxRXA

xA{Ww=�GgY[ANMTWxO0?:A

a_,O[MPagE:A

xa ≡n saWx=�ANGQ?:O ≡n

E:GQ;0MXACU�;[sd=�tjG�?BAksNOjG:v[MXEQANGQsN=�;LMPACWX_,A{s}RPO�?BA × `y ��[WWx=�KL_:U�ANKLANGgRXA[n

x = resto(s, n)nQ_6ANMPO>ACGHANU<sC;jWXOL?:A

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E:GQ;)?BAd],G:=�sN=lt[G�?BACUMXA{W�RPO³ACG A{WxAHsN;[WxO,` |k�r;VMT;�U@; E:GQ=�sN=�?:;[?bn�WXE:_,O[G:v[;³agE:AxA

yWxO[G�;jKkf6O[W>=lGJY[ANMTWxOgWk?BA

aANG

Zn`

��GgRXO[GQsdA{Wxa = 1

yq=�vjEQ;jU�?Q;j?HMPANU@;SRX=�YS;�;Zn| i ya = 1

n6OzWXAC;xa = ya

ANGZn`��EQU RP=l_:U�=@sN;VG,?BO�_6OjM

xnxax = xay

n�OWxA{;

(xa)y = (xa)xn,ODWxA{;

x = yANG

Zn`��2O[WWX=�vjE:=�ANGgRXA{WsdO[MXO[U�;jMX=�O[W�WxO[Gz=�G:KLAC?B=@;SRPO[WCh

� CAD?C &. DE:>C �B` ¸ 0 ��Zn 2��^¥T²N�Q�j²N¯65S�

ax = b�u¯4¥N�6¥�ªN¦S� �Q²N¯65S�mª}¯

a

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a 6= 0¥d�Zp

3� CAD?C &. DE:>C �:` ¸�P�0 ��

Zp²T¦S�

p © �}¯±¤D¦ 2 ª}¯ ab = 0¥N�Q� ¦S�6²T¥dª\¦

a = 0¦

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`�2+ �<+��^3 "�� " � 1 7:9�9{145 � � � � " � 7:9j+ JAC;jG

m, nsdO[_:MP=lKLO[WC`��GgRXO[GQsdA{WoWx=�ANKL_:MPADAdcB=@W�RPADE:GQ;�WXOjU�EQsd=�tjG

x?BADU@;jW

sdO[G:vjMPE:ANG,sd=@;jWWX=lKÊ:U RT;VG:A{;jWx ≡m a i x ≡n b

`Q��O[M�A��ACKL_:UlO,n,_,;VMT;x ≡2 1 i x ≡3 2

_:MPE:ANf6O0, 1, 2, 3, 4, 5 i YjACO�agE:A

x = 5sNE:KL_:UlA[` � A0e:A{sTe:OQn2YS;jK>OgWo;HYjACMoagE:A0_6OJ?:ANKLO[W\ACGQsdO[GgRXMT;VM\E:GQ;�WXOjU�EQsN=lt[G)_QMXO[fQ;VGQ?:OmU�O[WoGgIQK>ACMXOgW\?BACWP?BA��;

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�)��#� �?�� E��

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kagE:A\sdEQK>_QUlA

a + km ≡n bnBOLWxA{;ÊQG

kRP;VU

agE:Akm ≡n b− a

`J��;LagE:Am, nWxO[GzsdO[_:MP=lKLO[WCngANUb=�GQsdt[vjG:=lRXO

kWXAZ_:E:AC?:AZACGQsdO[GgRXMT;VM�_,O[M�ACU2sdO[MXO[U�;jMX=�O^�B` ¸ �B`[��U2GJI:KLANMPO

a + kmsNE:KL_:UlAÛ�;[W�?BOgW�sNOjG:v[MXEQANGQsN=�;[WN`:��OB?BACK>OgWwRPOjKD;VM

kANGgRPMXAk� i n − 1

n i km < nmn i _6OB?BANKLOgW�WXE:_6OjG:ACM�agE:A

0 ≤ a < m,;jWX�2a[EQA\_,OB?BACK>OgW�RXACG:ANM

0 ≤ a + km < nm`

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ZnsdE,;VGQ?BO

a, nWxO[G

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f(n)YS;H;

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n − 1)! ≡n n − 1ªT¯3LªO5V�l¦Lª}¯

n¥dª © �}¯±¤L¦ 2 ªN�S� °S¦m¥d��²T�Sªd¦ n = 4 3

4 ¥d¤D¦�ª}�u�X�j²N¯65S� 3 JE:_6OjGQv[;VKLOgWr_:MP=lKLANMPOoagE:A nG:O\A{W�_:MP=lKLO,`V��GgRXO[GQsdA{W

n = mqsNOjG

m, q > 1`[B=

m 6= qn[WX=lACGQ?BOk;jKkf6O[W

KLANG:O[M�agE:Ann[WXE0_:MPOJ?:EQs}RPOQn

nngACW�p4;jsdRXO[M�?BA

(n− 1)!n[;[Wx�6agE:A

(n− 1)! ≡n 0`BJ=

n = m2ng_:E:A{W�RPOkagE:A�eQANKLO[W�Adc:sdU�E:=@?BO

ANU�sN;[WxOn = 4

n,_6OB?BANKLOgW�;[WxACvjE:MT;VM�agE:Am i 2m

WXOjG ;jKkf6O[W�K>ACG:OjMPACW�agE:An i _6OjM�RP;jG[RPO0p4;[s}RXO[MXA{W�?BA (n − 1)!

n6agE:AeQ;[sdA\agE:A\_,O[M�RP;jG[RPODWxEm_:MPOJ?:EQs}RPO

2m2RT;VKkfQ=l�CG � U�E:ANv[O m2 = n

RP;jKkf:=��NG2n,;jWX�2a[EQAF?:A\GgEQANYjO(n − 1)! ≡n 0

`JE:_6OjG:vg;VKLO[W�agE:A

nACW�_:MX=�KLOQ`,��GgRXOjG,sdACW�sN;[?:;DANU�ANKLACG[RPO�WP;VU�YjOD�0?BA

ZnRX=�ANG:Ao=�GJYjACMPWXODI:G:=@sdO,`,�2OgWANU�ANKLACG[RPO[W�G:O

�m?BAZnWXOjG

1, ..., n − 1` ¸ ACWFWXE _:MPOj_Q=lOz=lGJYjACMPWXO i n − 1

y4agE:ALA{W −1ANG

Zn|�RT;VKkfQ=l�CG2` � =�G:vjI:G)OVRPMXOzACWFWXE _:MPOj_:=�O

=�GgY[ANMTWxO,nB_,O[MPagE:Ao?Q;j?BOa2 = 1

nQACWa2 − 1 = 0

nQUlE:ACvjO(a − 1)(a + 1) = 0 i _,O[M�sNOjMPOjU@;VMP=lOL�B` ¸%P nQO a − 1

Oa + 1

ACWw�QnACW�?BACsN=lM{n

aA{W

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ZnagE:A

(n − 1)! = n − 1`�

��

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` � ACKLO[WxRXMT;VM{h�g|

x ≡ yWX= i WXtjU�ODWx= [x] = [y]

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x, y ∈ UnBO

[x] = [y]O

[x] ∩ [y]A{W�YS;jsd��OQ`

²{| ⋃

x∈U

[x] = U`

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x, n := 3, 0; i MXAC_,ANRX=�M x, n := xˆ2KLOB?

(2ˆ32 + 1), n + 1;e,;jWxRP;�agE:A

YjA{;VKLO[Wn = 32

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x500 = 1000 i 5002 = 250000h�KL=�;jUlv[OjMP= RPKLO

sN;jU�sNE:U@;f(500)

ACGHK>ACG:O[W?:A ¸ �>WXANvjEQGQ?BO[WCnJKL=lACG[RPMP;[W " ACU2GJE:ANY[ODWxt[UlOLv[;jMP;jGgRX=�«C;êQ;jsNANMPUlO>ANG R e:O[MP;[WO$B��U<_:MP=lKLACMnACG

agE:A\ANU<GgEQANYjOL;VU�vjO[MX=lRXKLOLv[;jGQ;ÂCWn = 1000,

?:OjGQ?BA\U�O[W?:O[Ww;jUlv[OjMP= RPK>OgW�A{W�RT�VG�?BACK>O[MP;jGQ?BO>ANGgRXMPA ¸j¸ i ¸ �>?B��;[WN` � A{WX?BAU�E:ANv[OQnb?BAD;Ve:�rACG);[?BANU@;VGgRPAn2 vg;VGQ;zsdOjGgRPE:GQ?BACG[RPANKLANGgRPAjh ¸ � � � 2 = 1, 10

KL=�UlU�OjG:A{WNnx1050 = 1, 99

KL=lU�U�OjG:A{WNn ¸j¸ �j� 2 =¸ n�� ¸ KL=lU�U�OjG:A{WNnx1100 = 3, 98

K>=�U�UlO[G:ACWC`�wU�vjOD_Q;jMXA{sd=@?BOLOBsdE:MPMXAFsdO[GHANU<;jUlv[OjMP= RPK>OL�F&obh:G:ODANKL_:=�AN«{;D;>v[;VG,;VM;LUlOgW�YJ=lAx��O[Ww;jUlv[OjMP= RPK>OgW�eQ;jWxRP;>GJI:KLANMPO[W�RT;VG

vjMT;VG,?BACWagE:AFG:=�G:vjEQGQ;DsdO[K>_QEBRP;[?BOjMT;^sNOjGHG:=�G:vjIQGH;jUlv[OjMP= RPKLO^_:EQAC?BA\RPANMPKL=lGQ;jMw?BE:MT;VGgRXAFU@;LYJ=�?:;D?:ANU2O[_,ACMP;[?BOjM{`J��ACMXOLU@;e:=@W�RPOjMP=�;m?BADU�O[WFGJE:ACYjO[WF;jUlv[OjMP= RPK>OgW�A{WoagE:ALUlOgWF=lGJY[ACWxRX=�v[;j?:OjMPACW�UlOgW�KLAx��OjMT;VG2n i³i ;zeQ;VG)KLA��OjMT;j?:OmACU�;VU�vjO[MX=lRXKLOH�F&o_6OjME:Gmp4;jsdRXO[Mw?BAFKL�[W?BA\E:GHKL=lU�U�tjG2n:_6ANMPO>RXOB?:;�YJ�@;kU�AFp4;VUlRP;>OVRPMXO>p4;[s}RXO[M?BA ¸ �[�j�j�[�o_Q;jMP;>WXANMwsdO[KL_,ANRX=lRX=�YjOQ`��; i KkEQsTe,;jW0;j_:Ul=@sN;[sd=�OjG:A{WL_Q;VMT;)_:MP=lKLO[W0vjMT;VGQ?BA{WNn i Y[ANMPANKLO[WLE:G,;ÁNG U@;8sdMP=l_:RXOjv[MP;Vpq��;)KD�jW�;j?BACU�;jG[RPAj`��ACMXO¡U@;f:IQWPagE:AC?Q;m?:AD_:MX=�KLO[WFKD�jW i KD�jWFvjMT;VG,?BACWFeQ;HWX=�?:OmEQG³],G¡ANG´Wx��KL=@WxKLO,n2E:GQ;HsdO[KL_,ANRX=@sd=�tjGÁNGgRXMPALKD;SRXACKD�SRX=@sdOgW\agE:A

i ;0ACWxRP;Vf,;DANG�_:U�ANGQOz?BA{WX;jMXMPOjU�UlOLACG�ANU�Wx=�vjU�O ¸ �L_,ACMXO�agE:AÂdcB=@W�RP��;DACGgRXMPAoUlOgWwv[MX=�ANv[O[Ww;jG[RP=lv[E:O[W�sdOjKLO0ACYJ=�?BACGQsd=@;DU@;�sNMX=�fQ;?BAF��MT;SRPt[WxRXANGQACWC`[��ANMPO ��ANMTWxACG:G:A i U:ACMXKD;SR�ANG,sdOjGgRPMP;jMXO[GD_:MX=�KLO[W�agE:AZG:OL_,OB?B�@;VGmWxO Q;VM�UlOgW�vjMP=lACvjO[WCn i sdOjGgRP=lGJEQ;jK>ACGgRXAANG)ACWxRXOgW � WX=lv[UlOgW�WXA>eQ;�ACGQsdO[GgRXMT;j?BO,n6WXOjf:MPA>UlOgWwRPMP;jfQ;S��OgW�?BALACUlU�O[WCn�_:MX=�KLO[WF?BA>KL=lU�UlO[G:ACWF?BAD?B��vj=lRXOgWNnbWX=lACGQ?BOmANUrI:UlRX=�KLO?BA{WXsNE:f:=�ANMXRXOHANG)U�;H�G:=lY[ANMTWx=@?:;[?³?BA��;jUl=lpqOjMPG:=@;�ACG´�<O[WF�wGQvjANU�ACWCnbANU�� ¹ ?BA0�wv[O[WxRXOH?BA��V�[� P h,RX=�ANGQA ¸ � R � P0R;P;R ?:�lv[= RPO[WCn_6ANMPO^RP=lACG:A\E:GQ;>ANcJ_QMXA{Wx=�tjGmKD�jWsdO[KL_Q;jsdRP;:h

243112609 − 1`

��G8U�O[W\IQU RP=lKLO[W^;� :OgWNnbANcB=�WxRX=�t�E:GQ;�KLOjRX=�Y�;[sd=�tjG´GJE:ANYS;H_Q;jMP;HfQEQWXsC;VM\_:MP=�K>OgW\vjMT;VG,?BACWChbeQ;VfQ��;HE:G¡_QMXACK>=�O�_Q;jMP;HACUagE:AFANGQsNE:ANGgRPMXAFANU<_:MX=�KLANM_:MP=�K>OD?BA ¸ �^KL=�UlU�OjGQACW?BA\?B��vj=lRXOgW�O>KD�[WN`:�<O[Wsd=�ANGgRP� ],sNO[W?BA\F��<�$U�O>vg;VGQ;jMXO[G2n i ;VeQOjMT;kACU?BA{WX;Vpq�lO0WXA\MXACG:OSYjt,hJv[;VG,;VMT�>sN=lACGgRXO0sd=�GQsNE:ANGgRP;LKL=�U�?Bt[U�;jMXA{W?BA\_:MPANKL=�O0;jagE:ANU�a[EQAoANGQsNE:ANGgRPMXA\ACU2_:MP=lKLANM�_:MX=�KLO0sdO[GmKD�[W?BAosN=lACGzKL=�UlU�OjG:A{Ww?BA\?B��vj=lRXOgWN`J��;DagE:AFACWxRP;jKLO[W�ANGzRXMPACsNAFK>=�U�UlO[G:ACWCn:G:OLeQ; i agE:AF_6ANMT?BANM�RP=lACKL_,O,`��G ¸ �.P;P nJANU<_QMX=�KLODKL�[W�vjMT;VGQ?:A\_:E:f:U�=�sC;j?BODACMP;L?BA D?B�lv[= RPO[WCn:WX;jUl=�ANG,?BO0?BO[WwANGHACWXAo;� :O,n

217 − 1 i 219 − 1.��G�ANU

WX=lv[UlO ¸ � ��ANMTWxACG:G:A�sNOjGS��ANRXE:MPt>U�;k_:MX=�KD;VU�=�?Q;j?0?BAZYS;VMP=lOgW�_QMX=�KLO[W�?BA�A{WxA�RX=�_6O2p − 1

sdOjGp_:MP=�K>O,nBa[EQA�WXA�U�U�;jKD;VGz;je:OjMT;

© �}¯±¤D¦�ª\¨j¥�� ¥N�Tªd¥N�Q��¥dn[_,ACMXOkG:Ok?BACK>OgW�RPMXt\GQ=lG:v[E:G:O i WxA�ACagE:=�YjOBsdt\ACGDY�;jMX=�O[WC`V��EQUlACM�?BACK>OgW�RPMXtoagE:A 231 −1y ¸ �\?:�lv[= RPO[WT|

ANMT;\_:MP=�K>O,n i ANG ¸%P ��ZWX;jUl=�toE:GQO^?BA ¸C¹ ?B��vj=lRXOgWN`[��G ¸�P � \�2EQsC;jW�ACGQsdO[GgRXMPtoANU6_:MP=lKLO^� 127 − 1n[agE:A�_QMXO[fQ;Vf:U�ANKLACG[RPAwACW�ANU

� � ��

_:MP=lKLOFKD�jW�vjMT;VG,?BA�?:ACWPsdE:f:=�ANMXRXO\sNOjGLWxt[UlOF_Q;j_,ACU i U@�V_:=�«jn i a[EQA�?:E:MXt\sNOjKLO\sN;VKL_6ANt[GêQ;[W�RT; ¸%R�:¸ sNEQ;VG,?BO (2148 +1)/17y �� ?B��vj=lRXOgWP|�WP;VU�=lt,`C��G[RPMP;ZACGÂCUBACWPsdANG,;VMP=lO�ANG>ACWXA�;! :OFU�;ZsdO[K>_QEBRP;[?BOjMT;w_:MPOjv[MP;jKD;Vf:U�Ajn i GJE:ANY[O[W�sN;jK>_6ANO[G:ACW�ACK>_6AN«{;VfQ;jG;HWX;jUl=�MFE:G:OmRXMT;jWZOVRPMXO,nbsd=�GQsdOHANG ¸%R1� �:n6OjRXMPOzACG ¸�R� �Jn�sdEQ;VRXMPOm_6OjM\?B�{sN;[?:;mANGgRXMPA ¸�R [� i ¸�R;R �:n�Wx=�AdRPA>ACG³U�O[W R �:n i i ;OBsTe:OzKD�[W\?BACWP?BAD�V�j�[�:`��UrGJI:KLANMPO�?BAD?B��vj=lRXO[WF?BACUr_:MP=lKLO�sN;jK>_6ANt[G³e,;zsNMXA{sd=@?BOm;mE:G)_:MPOjKLAC?:=lOH?BA ¸ � P �[�j�D_6OjM\;� QO?BACWP?BA ¸%R1�B¸ yqACU2vjMT�S],sNOkA{WwsN;[Wx=bU�=lGQAC;VUunBYjACMANU2WX= RP=lO��ANf0�X��e:AF��MX=�KLA\�r;VvjA{W �x|}`

� AzACWxRXO[W ¹ � sC;VKL_6ANOjGQACWCn ¹ ��eQ;VG8WX=�?:O�_:MP=lKLO[W^?BA ��ANMTWxACG:G:Aj`�wG8GJI:KLACMXO ?:A ��ANMTWXANG:G:A�ACW>Wx=�ANKL_:MPA0_6OVRXACGQsd=@;VUsN;jGQ?B=@?:;SRPO>_6OjMACU2Wx=�vjEQ=lACG[RPAB�+�CADE+�-/. `JwGzGJI:KLANMPO

2n − 1ACW�_:MX=�KLOkWX= i WXtjU�OkWx= n ACW�_:MP=lKLO^=lKL_Q;jM i Ln−2 = 0

?BOjGQ?:ALA{W�U@;oWXEQsdA{Wx=�tjGz?BAd]QGQ=�?:;

_,O[ML0 = 4, Lm+1 = L2

m − 2K>OB?

(2n − 1)`

�<;F?BANKLOgW�RPMP;[sd=�tjGL?BA��CWxRXO\EQWP;FWxt[UlOF_:MP=lG,sd=�_:=lOgWrWx=�KL_:U�ACWr?BAU@;ZRXACOjMP��;Z?BAGJI:KLANMPO[WCnVsdO[K>OFe:ACKLO[W�YJ=�WxRXO,nS_,ACMXO\KÊQsTe:OgWANG¡sN;[?BANG,;:nbagE:ADOBsdE:_Q;jG RPMXA{W\_Q�jvj=�GQ;jWZANG &FGJEBRPe2nbRXO[KLO��:n i G:OHU�O�MXAC_:MPOJ?:EQsd=�KLO[WF;jagE:�unbAdc:sdAC_BRXOH_Q;jMP;H?BA{sd=�MoagE:A0Wx=nGQO�ACW\_:MP=�K>O,n

n = uvsdO[G

1 < u, vn2ACG[RPOjGQsNACW\_6OjMkU�;H=@?BANGgRP=�?:;[?

xk − 1 = (x − 1)(1 + x + x2 + ... + xk−1)ACW

(2u)v − 1 = (2u − 1)(1 + 2u + ... + 2u(v−1))n<A{Wk?BA{sd=�M

2uv − 1ACW^?B=�Yg=@WX=lf:U�A0_6OjM

2u − 1`<��OjMkA��ANKL_:U�OQn

27 − 1ACW

_:MP=lKLOk_6OjMTagE:AL0, L1, L2, L3, L4, L5 = 4, 14, 67, 42, 111, 0

`C�$�CWxRXAZACW�ANU�;VU�vjO[MX=lRXKLOQng=�G[RPMXOB?BE,sd=@?BOk_6OjM��2EQsC;jWZy ¸�P � g| iKLA��OjMT;j?:O^_6OjMw�<ANe:KLANMoy ¸%R[¹ �[|dnBa[EQAoWxAFEQWP;L;Ve:O[MP;^_Q;VMT;^f:EQWPsN;VM_QMX=�KLO[W�vjMT;VG,?BACWC`��Ub_:E:GgRXOD?BAF_Q;jMxRP=�?Q;o_Q;jMP;ÂNUb;VU�vjO[MX=lRXKLO>�F&oDA{W�E:GQ;^vjACG:ANMT;VU�=l«{;jsN=lt[G0?BANU�RPANOjMPANKD;^?BA�U:ANMPKD;SR;k_6OjU�=�G:OjKL=�O[WChJWX=

nACWsNOj_:MP=lKLOLsdO[GaANGgRXO[GQsdA{W

nACW_QMX=�KLOLWX= i WXtjU�OLWx=

(x − a)n ≡n (xn − a)_Q;VMT;kRXOB?BODGQ;VRXE:MT;VU

xn

UlO�sNEQ;VU�A{WFACagE:=�YS;VU�ANGgRXA0;H?BA{sd=�M�yqY[ANM\A��ACMPsN=�sN=lOJ|Za[EQA0Wx=�KL_:Ul=l],sC;VGQ?BOHU�O[W\?BOgW\_6OjU�=lG:O[KL=lOgWFUlOHKD�jWF_6O[WX=lfQUlALRPOjKD;VGQ?:OmU�O[WsdOJAd]6sd=�ANGgRXA{WbKLOJ?

nn i ;j_:MXOSY[ACsTeQ;jGQ?BO xn ≡n x

_Q;VMT;�RXOB?BOx, nnCUlOgW2?BO[W<_,O[Ul=�G:O[K>=�O[W<WxO[GZ=�vjE,;VU�ACWC`d��W2EQGQ;wsN;jMP;[s}RPANMP=l«{;jsd=�tjG

ANU�ANvg;VGgRXAF?BA\GJI:KLANMPO[W�_:MP=lKLOgW�_,ACMXOLG:ODKÊ i IBRP=lU2_,;VMT;>sC�VU@sdE:U�O[W�_6OjMTagE:AFANU<sC�VU@sdE:U�O>?:A0y x − a)nACWKÊ i U@;VMPvjOQ`J��ANMPODWx=WxAk_:EQ?B=�ANMT;LRXMT;Vf,;S�x;VM�sdOjG�U�O[W�MXA{W�RPO[WKLtB?BE:U�O�EQG�_,O[Ul=�G:O[K>=�O�sNOjKLO

xr − 1sdOjG

rWXEB],sN=lACGgRXANKLACG[RPA^sTe:=@sdO,nQACU�sN�jU�sNE:U�O�WxA

;jsTe:=@sN;Q`�we:O[MP;Qn:Wx=

nACW�_:MP=�K>O i Wx= r i n

WXOjGHsdO[_:MX=�KLO[WsNOjGaANGgRXO[GQsdA{W

(x − a)n ≡ xn − 1KLOB?

(xr − 1, n)?BOjG,?BAp(x) ≡ q(x)

KLOB?(s(x), n)

agE:=�ANMPA\?BACsN=lMwagE:AFRPOjKD;VGQ?:O>U�O[WsNOJAd],sN=lACG[RPACW�KLOB?nn:U�O[W�MPACWxRXOgW�?:A

p, q?B=�Yg=@?B=@?BO[W�_6OjM

sWXOjGm=lv[EQ;VU�ACWC`Q�C��;jMP;LsNEQ�VU�ACW

a, rWxAF_:E:A{?BAo?BA{sdU@;VMT;VM�U�;LMPACsN�l_:MPOBsN;g�F�<;>MXA{Wx_QE:ACWxRP;ÛlU�ANv[tDsdOjGmANU2WX=�vjE:=�ANGgRXA

B�+�CADE+�-/. `by �wvjMT;��;VUun!&\; i ;jU2nJ:;ScBANGQ;g|}`BJAC; s ≤ n`BJE:_6OjGQv[;kagE:A�UlOgW�_QMX=�KLO[W

q, rsdE:KL_:U�ANG2h

q | r− 1nn(r−1)q 6= 0, 1KLOJ?

rn i

(

q + s − 1s

)

≥ n2b√rc `J=2_Q;jMP;kRXOB?BO

1 ≤ a < sRXANGQANKLO[W

yq=±|aACWsNOj_:MP=�K>ODsNOjG

nn iyq=�=@|

(x − a)n ≡ xn − aKLOJ?

(xr − 1, n)ANGgRXO[GQsdA{Wn = pk _Q;VMT;L;VU�vjI:Gm_:MP=lKLO p i k > 1.�<;0ANcB=�WxRXACGQsd=@;0?BA

q, r, sagE:A^sNE:KL_:U@;VG�U@;0e:=�_6tVRXA{Wx=@WZ?BAkA{W�RPAoRXACOjMPANKD;DO[MX=�vj=�GQ;VU�KLANGgRXAkMXA{a[EQ=lMP=ltDACU�E,WxOm?BAkU@;DRPANOjMP�@;

;VGQ;jUl�lRX=@sN; ?BA�GJI:KLANMPO[WCn i � &o�n�G:O³WX=lGgRP=l�CGQ?BO[WXAzWP;SRP=�WxpqACsTeQO[WksNOjG WXE8_:MPOj_Q=�;�?:ANKLO[WxRXMT;jsN=lt[G8Wx=�G8ANGgRPANGQ?BACM>;�pqOjGQ?:O ANU_Q;jWXODagE:Ao;V_6ANU�tD;LACWP;kRPANO[MX�@;:n:WXAFMX=�GQ?B=�ANMPOjGmANGHANU<=lGgRXACGgRXOQn,?B=�sN=lACGQ?BO0��vjMT;��;jU6agE:A\U@;>RXACOjMP��;L;VG,;VU�� RP=�sC;>?:A\GgIQK>ACMXOgW�ANMT;?BANKD;[Wx=@;j?BOL?BACGQWX;Q` � A{Wx_QE:�CWwWP;VU�=�tÊ:G,;L_,A{a[EQAO Q;>YS;VMP=�;[sd=�tjGm?BANUbRPANO[MXACKL;>agE:A\_6ANMPK>=lRX=�tLANU�=lKL=�GQ;VMACU2EQWXOD?BAFACWP;kRPANO[MX�@;:`��UbRXACOjMPANKD;:n i UlOgW�Ul=�KL= RPACWA{W�RT;Vf:U�ACsN=�?BOgW�_,O[MUlOgW q, r, s sdOjG,?BES��O0?B=�MXA{s}RT;VKLANGgRXAF;VU�;VU�vjOjMP=lRXKLO^� &o�h

primo(n) =9C1 (n = ab

_Q;VMT;^;jUlv[I:Gn > a, b > 0

) "�39C1!"�3��7 � 7

r� � 9 � � �� 7:9 �C7

n� 7 �67

9C1(mcd(n, r) 6= 1)

"�39C1!"�39{1

(rA{W�_:MP=lKLOLKD; i OjMagE:Ao�j|

q :=ANUbp4;[s}RXO[M�KL�[W�vjMT;VGQ?:AF?:A

r − 19{1(q > 4

√r log n � n(r−1)q G:OLACW ≡r 1)

� �� 7��9C1 y � /2149 � �

an1 ≤ a ≤ 2

√

(r) log nnJRT;VU<agE:A

(x − a)n 6= (xn − a)KLOB?

(xr − 1, n))"�3

9C1!"�3´9��W_6OjU�=lGQtjKL=�sNOL_:MXO[fQ;VMWX=

nA{WE:GQ;>_6OVRXACGQsd=@;:n i U�;L?BACKLO[WxRXMT;jsd=�tjGzACW�ACUlACK>ACGgRP;VU<_,ACMXOD?BAFYS;VMP=�;[W�_Q�Vv[=lGQ;[WN`

*��<+�� 1 � �C3 � � 7 � � 7JE:_6OjG:vg;Da[EQA\ANU<;jvjANGgRPAF�[��âgE:=�ANMPA\MXA{sd=�f:=lMKLACGQWX;V��ACW?BACU<;Vv[ANGgRXAo� ¸j¸ _,O[M�sdO[MXMPANO>ANU�ACsdRXMPtjG:=@sdOL_6ANMPOLRXACK>AoagE:A\U�O[W

KLANGQWP;S��A{W�_:E:AC?:ANG0WXANMr=�G[RPANMTsdAC_BRP;[?BO[W�_6OjMrACUQeQ;[s)(jANM�ACG:ANKL=�vjO , ;jWX_:EBRPG:= (6`[�2O[W�K>ACGQWP;S��ACW�GQO\MXA{a[EQ=lACMXACG ��=@sdMPO[WXOVp±R��)OjMT?G:=:��AdcF¬\_:E:MPOZ�ZQ� ~x~ nVsNOjGkEQG>;jU p4;jf,ANRXOF?BA�RT;VKD;� QO

28n�fQ;jWxRP;Q`S�we:O[MP;Qn{EQG^KLANG,WX;V��A�?BA

ksC;VMT;jsdRXANMPACW�?BA��Q� ~X~ WXA�_QE:AC?BA

� �Q� ��#� ��?��E�?� �� PºsdO[GQWX=�?BACMP;jM�E:GzGgIQK>ACMXOkACWPsdMP= RPOoANG0U@;of,;jWXA

28,O^WxA{;oE:G0GJI:KLACMXOÂCWPsdMP= RPOoANG0fQ=lGQ;jMX=�Ok?BA

8k?B��vj=lRXOgW�f:=�GQ;jMX=�O[W�yl�xf:= RTW ��_6OjM

ANU,=lGQvjU��CW��RN=�GQ;VM i ?B=�vQ¯±� ª}�x|}`[��GgRXOjG,sdACW�sN;[?:;\KLANGQWP;S��AA{W�E:G�GgIQK>ACMXO,n i �j��ZagE:=�ANMPA�agE:AwACUQGJI:KLANMPOkWXAC;\sNOJ?:= ],sC;j?BOoWXANv[I:GE:GQ;msdU@;�YjA>agE:A , ;jWX_:EBRXGQ= (�G:OmsdO[G:OBsdAj`b�G³WX=lKL_:U�A^KD;j_,ACOzf:= i A{s}RP=lY[Oz?BA>UlOgWZGgIQK>ACMXOgWwKLANGQOjMPACWZa[EQA 28 ACG³WX��KL=@WxKLOgWG:OHfQ;[W�RT;:nb_6OjMTa[EQAÂ{WZp4�jsd=�U�?BALMPOjKL_6ANM\ANG)fQ;[WxAD?BALU@;jWZpqMPACsdEQANGQsN=�;[WZsNOjG:OBsN=�?:;[WF?BADU�;[WZUlANRXMT;jWFANG´sN;[W�RPANU�U�;jG:OQ`b�?BACKD�jWCnagE:=�ANMPA�sN;jKkf:=@;VM�ANUbsNtJ?:=lv[O^?BAZYjAC«�ACGzsdE,;VGQ?BO i agE:=lACMXAZ=�GQ?B=@sN;VM�_6OjM�sdOjMPMPANOoACUlA{s}RXMPtjGQ=�sNO^sdE,;VUbsdtB?B=�vjO>a[EQ=lACMXAZa[EQAZWXA�E,WxA[`��GgRXO[GQsdA{W�UlAZvjEQWxRP;jMX�@;\A{Wx_6ACsN= ],sC;VM�_,O[MsdOjMPMPANOoACUlA{s}RXMPtjGQ=�sNOkagE:AZsC;j?:;^KLANGQWP;S��A

MWXAC;ksNOB?B= ]6sN;j?:O^sdO[KLO

f(M) � ?BA{Wx_QE:�CW�j��ANG�WXE�sC;jWP;\YS;o;k;j_:Ul=@sN;jMXU�Af−1_Q;VMT;\ACG[RPANGQ?:ANM�ACU,KLACGQWX;V��Aj`[��ACMXO^?BANf6ANMP��;^WxACM�?B=lpq�@sd=�U,_Q;jMP; , ;jWX_:EBRPG:= (DsC;VU@sdE:U@;VM

f−1;VIQGmsNOjG:OBsd=�ANG,?BOf`:�wU2KL=@WxKLOLRX=�ANKL_6OQn:GQOL?:ANf6ANMP��;LWXANMw?B=lpq��sN=lU<_Q;VMT;^� ¸j¸ sN;jU�sNE:U�;jM

f(M)`

��OjM\A��ACKL_:UlO,n<JAC;pE:G´_:MX=�KLOzv[MP;jGQ?BA[n i a

E:GÁNU�ANKLANGgRPOm_QMX=�KL= RP=lY[Om?BAZp`�BAC;

f(M) = aMKLOB?

p`fACWFE:G,;

f:= i A{sNsd=�tjG>?BAZpACGLWX�BKL=�WXKLOQ`V��;Z?:= ],sNE:UlRP;j?LsNOjG>ACWxRXOZACW�agE:A

f−1YS;F;FWxACM�RP;VGL?B=lpq��sN=lUQ_Q;VMT;��[��sdO[K>OF_Q;jMP; , ;[Wx_:E:RXG:= (�`

��U@;VMT;VKLANGgRXAFWX=p = 2232582657 − 1

nBGQODY�;L;LWXANMwWP;SRX=@Wxp4;js}RPOjMP=lO>KkEQU RP=l_:U�=@sN;VM_6OjMaeQ;jWxRP;LagE:Ao;V_,;VMPAN«CsC;

f(M)`

�RXMPOoA��ACKL_:UlO,hgWXAC;f(M) = M2

KLOJ?n`g��e:OjMT;FANU6MXA{sd=�_:=�ANGgRXA�?:A

k = f(M)RX=�ANG:AZa[EQA�sN;jU�sNE:U�;jM √

kACG

Zn`[��W�RPOkA{W

?B=lpq��sN=lU�_Q;jMP;\;jUlv[E:G:OgWnng_,ACMXOoGQOkWx=

nACW�E:G0_QMX=�KLOoACagE:=�YS;VU�ANGgRXA�K>OB? � ; ¹ h

n = 4j +3`BJANv[I:GDANU6_,A{agE:AO :OkRXACOjMPANKD;o?BA

U:ACMXKD;VRCnk4j+3 = k

KLOJ?(4j + 3)

n:ANGgRXO[GQsdA{Wk4(j+1) = k2

n:ACG[RPOjGQsNACWk2(j+1) = ±k

nB_6ANMPOLG:OL_:E:AC?:AFWXANM −k_,O[MPagE:AFACG

ACWXAosN;[WxOLRPANGQ?:MX�@;VKLO[WwE:GM2 = k i EQG x2 = −k

ACGZnnQACGgRXOjG,sdACW

(Mx )2 = −1

n:_6ANMPODACWxRXAZnG:O0_:EQAC?BA\RPANG:ACM √−1yqY[ANM�A��d`b�B` �J` ¸ �[|dn:OzWXAC;

(kj+1)2 = kn i sdOjG,sdU�E:=lKLOgW�agE:A M = ±kj+1 = ±k(n+1)/4 KLOB? n nQ_,;VMT;DU�;�?BACsNOJ?:= ],sC;jsN=lt[G�?BA

f (M)n i WxACvjE:MPOLa[EQAZEQG:O0?BAFACWxRXOgW�G:O>RX=�ANG:AoWXANGgRX=@?BO,`B��ACMXOLA{W�RPO>A{W�p4�jsN=lU2_,;VMT; , ;[Wx_QEBRXG:= (0RT;VK^f:=l�CG2`*��<+4*6+ �� 5��{3 � 3 � �� 7 ��1+"�+ ��ANMPODWxE:_6OjGQv[;kagE:A

n = pq?BO[GQ?BAF;VKkf6O[W�?BA

p i qWXOjGzACagE:=�YS;VU�ANGgRXA{W; ¹ KLOB?

� `��weQOjMT;�� ¸[¸ KD;VGQ?Q;k = M2 KLOB? pq

`��WF?BA{sd=�MCnM2 = jpq + k

n�;[Wx��agE:AM2 ≡p k i M2 ≡q k

n i _,O[M�RT;VGgRXO)yqYjACMA��ACMPsN=�sN=lO ¸ |

M ≡p ±kp+1

4i M ≡q ±k

q+1

4

`gJA�_:E:AC?:ANGD;VMPKD;VM�ANGgRXO[GQsdA{W � Wx=@W�RPANKD;jWr?:A�?:O[WrsdO[G:vjMPE:ACGQsd=@;jW�WX=lKÊ:U RT�VG:A{;jWCnANGgRPMXA\sdE i ;jW�WXOjU�EQsd=�OjG:A{W�?BANf6A\ANU�ANv[=lM�j��J`g��Ww?BA{sd=�MCn:�[��FRX=�ANG:AoagE:AFANU�ANv[=lM�ACG[RPMXA�xMXACE:G:=�tjG�A{W�RT;>G:OBsTe:A\;kU@;jW ¸ �kACGzU�OD?BA� ¸{¹ �}ng� � � f �� f:E:f:E:f,;��}nJ��8$ «NcBYgf<n � ¹ � � ?� w� i �XWXANM � O � G:O � WxACMCnBA{WxOmACWZU@;0vj«COjMPG:ANGQ_:U�;VRX«d�T` � OzRXACGQ?BMT��?B=l],sdEQU RT;j?³ANGWP;Vf6ANM�sdEQ�jUJ?:A��{W�RPO[W�ACW�ANUJKLANG,WX;V��Aj` , ;jWX_:EBRPG:= (6n�ACG>sC;VKkfQ=lO,nCG:OFWP;Vf6A

p i qnV;VI:G>sdOjGQOJsN=lACGQ?BO�ANUJKL�dRXOB?BOF?BAsdOB?B=l],sC;jsd=�tjG<`

��WxRXA\KL�dRPOB?BOLACWw?BACf:=@?BOD; , ;Vf:=�G2`��ANMPOQn6� , ;[Wx_QEBRXG:= (mRX=�ANG:A^G:ACsNACWP;VMP=�;jKLANGgRXAoagE:AêQ;jsNANM�U@;Lp4;[s}RXO[MX=�«C;[sd=�tjGH_Q;jMP;DWP;jsC;VMwA{WX;[WwMT;V�@sdACW�sdEQ;[?BMP;[?:;jWwANG

Zn�

�2OágE:Am_,OB?BACK>OgWL?BANKLO[WxRXMT;VMÂCWLagE:AHWX;[sN;VG,?BO A{WX;[WkMT;V�@sdA{W^sNEQ;j?BMT;j?Q;jWCn�sdO[G E:G�_,OBsNO³KD�jW^RPMP;jfQ;S��O WP;jsC;VKLO[Wp i q

`��OjMTagE:Ajn�WXE:_6OjG:vg;VKLO[WLagE:A

MRP=lACG:AHU�;[WDsdEQ;VRXMPO´MP;j��sNACWLsdEQ;[?BMT;j?:;[W>?:=�WxRX=�GgRP;jWC`r��OjMDU�O¡a[EQA�e:ANKLO[WLYJ=@W�RPOéQ;jsNAm_6OBsdO,n

WP;Vf6ANKLO[W�a[EQAkRP=lACG:ANG agE:ALWXANM ±a,±b_Q;jMP;�;VU�vjI:G

a, b`6��e:OjMT;:n

a2 = b2ACG

Zpqn6_,O[M�RT;VGgRXO

a2 − b2 = 0ANG

Zpqn6_,O[M

RP;jGgRXO(a− b)(a + b) = 0

ANGZpq` � =lGQvjE:G:OL?BAZACWXO[W�p4;jsdRXOjMPACW�ACW��ÂNG

ZpqnJANGgRXO[GQsdA{W�_,O[M�U�O^KLACG:O[W�E:G:O>G:O>ACW�sdO[_:MP=lKLO

sdO[Gpqyst = kpq =⇒ p|s O p|t |d`2JE:_6OjG:ACK>OgW�agE:A p|a − b

`��GgRXOjG,sdACWqG:OH?B=�YJ=�?BAL;

a − by4_,O[MPagE:A^=lKL_:U�=@sN;VMP�@;�agE:A

pq|a − bOzWxA{;0agE:A

a − b = 0ANG

Zpq|d`,��GgRPOjGQsNACW�RXANGQANKLO[W�a[EQA

p = mcd(a − b, pq)n imi ;0agE:A q

G:Oz?B=lYJ=@?BA^;a − bn

q|a+b i sdO[GQsdU�E:=�KLO[WragE:A q = mcd(a+b, pq)`S��G[RPOjGQsNACWCnjWP;jsN;jGQ?BO

a, b, ;jWX_:EBRPG:= (^WXtjU�OZGQACsdA{Wx=lRP;FWP;jsC;VM

mcd(a−b, n) imcd(a + b, n)

_Q;VMT;kWP;Vf6ANMp i q`BB;jsC;VM

mcdA{W�p4�[sd=�U ngANGgRPOjGQsNACW�Wx= , ;[Wx_:E:RXG:= (>RX=�ANG:AZE:G�KL�NRXOB?BO>sNOjG�agE:AZACW�p4�jsN=lUbWP;jsC;VM

U@;jW�MP;j��sNACWCn[U�AZACW�p4�jsN=lU6p4;[s}RXO[MX=�«C;jMpq`BBANvjIQG��m¯��S¯ © ¥T¨S¯4�Sn,ANGmANUb;VMXRX=@sdEQUlOLWxO[f:MPA�sdMP=�_BRXO[vjMT;Spq�@;:nj�{W�RPAFACW�ANU�I:G:=@sdO>KL�dRPOJ?:O>ACGagE:AFU�;D?:ACsdOB?B=l],sC;jsd=�tjGzACWw?:ANKLO[WxRXMT;Vf:U�ANKLACG[RPA�RT;VG�?B�l],sd=�U<sNOjKLOÛ@;>p4;jsdRXO[MX=�«C;[sd=�tjG2`

*��<+ �2+ �� 5��C3 � 3 �� - + ��UBK>�NRXOB?BO , B�knV_,O[MrWxEQW�=�GJYjANGgRPOjMPACW , ` , =�YjACWxRCnS�k`jJeQ;VKL=�M i ��`S�?BU�ANKD;VG<n�RP;VK^f:=��NGANKL_:U�AC;H;jMX=lRXKL�dRP=�sC;mK>OB?BEQUlOn = pq

?BOjG,?BAp i q

WXOjG)_:MX=�KLO[W\?B=@WxRX=�G[RPO[WC`2��;jKkf:=��NG³]Qv[E:MP;jG³?BOgWFGJI:KLANMPO[Wd i e

agE:AACWX_6ACsd=l],sC;VMPANKLO[W�ANG�WXANv[E:=�?Q;:`B�j��\KD;VG,?:;>_,O[MwsdOjMPMPANOL;>� ¸j¸ U�;>pqEQGQsd=�tjG

f_,;VMT;kU@;LsdOB?B=l],sN;[sd=�tjGH?BANUbKLANG,WX;V��A

Mh

f(M) = M e KLOB? n `��WX�<agE:AFKD;VGQ?Q;^_6OjMYJ�@;>_:I:f:U�=�sC;n i en:agE:A\_:EQAC?BACGHWXANM�=�GgRXANMTsdAC_BRP;[?BO[W�_6OjM , ;jWX_:EBRXGQ= (�`:�j��o?:ACsdOB?B=l],sC;>sNOjG

f−1(m) = md KLOB? n `�r;VMT;�WX;jf,ACM^sNEQ�VU�ACW^WxO[Ge i d

n�G:A{sdA{Wx=lRP;jK>OgW\OVRPMXO�RXANO[MXACKD;:`<�GQ;�v[ANG:ACMP;jUl=�«C;[sd=�tjG´?BANU�RXACOjMPANKD;�_6ACagE:AO QO ?BA UQANMPKD;SRC`��MP=lKLACMXOmG:OVRT;VKLO[WFagE:AL_Q;jMP;msdEQ;jU�agE:=�ANM

nnbU�O[WZGJI:KLANMPO[WF?BA {1, ..., n} sdO[_:MX=�KLO[WFsdO[G n

pqOjMPKD;VG EQG³v[MXE:_6OHfQ;S��OmKkEQU �RX=�_:U�=�sC;jsN=lt[G�KLOJ?

nh�WX=

ab = kn + r i rGQO�ACWZsdO[_:MP=lKLOzsNOjG

nACGgRXOjG,sdACW

kn + rACWZKkIQU RP=l_:U�Oz?:A>;jUlv[I:G�_:MP=lKLOm?B=lYJ=@WxO[M

?BAnn i ACWXA\_:MP=lKLOLRPANGQ?BMP�@;Da[EQAk?B=�Yg=@?B=�MwO0; a

O0;bnQ;jWX�<agE:A

a i bG:O0WxO[G�;VKkf6O[W�sdOj_QMX=�KLO[WsdO[G

n` � ;[?BO

asdOj_QMX=�KLO

sdO[Gnn i ;0WP;Vf6ANKLO[WZagE:Ak_6OB?BANKLOgW�MPACWXOjU�YjACM ax = 1

KLOJ?nG:OVRT;VGQ?BOza[EQAkAdcB=@W�RPANG

r, sRP;jUlA{W�agE:A

ra + sn = 1n6;[Wx��agE:A

rKLOJ?

nWXANMP��;>?B=@sTe:OL=lGJY[ANMTWxO,`[��U<OjMT?BANGH?BAFACWXAFvjMPE:_,O>RP=lACG:AFANUbG:O[Kkf:MPAFACWX_,A{sd=@;VU

ϕ(n)`��OjKLOÂCGmRXOB?BOLvjMPE:_6OQnJANU2O[MP?:ANG

?BAosNEQ;VU@agE:=lACM�ANU�ANKLANGgRXOD?B=�YJ=�?:Ao;VUbOjMT?BACGm?BACU2vjMPE:_6OQnBACG[RPOjGQsNACW�ANU<GgEQANYjO^RXACOjMPANKD;>ACWJ=

aACWsdO[_:MP=lKLOLsdOjG

nACGgRXOjG,sdACW

aϕ(n) = 1KLOJ?

n`

��OjG,WxA{sdE:ACG[RPANKLANGgRPA�_Q;jMP;asNOj_:MP=lKLO�sNOjG

nRPANG:ACK>OgW

aϕ(n)+1 = aKLOB?

n`V��e:OjMT;:n�ANU

na[EQA�GQO[W�=�G[RPANMPACWP;�;[a[EQ�BACW

n = pqn

?BO[GQ?BAjnrsNOjKLO³?B= ��=�K>OgWNnp i q

WXOjG ;jKkf6O[W>_:MP=lKLO[W i ?B=@W�RP=lGgRPO[WC`��;jKLO[W>; Y[ANMLa[EQAzACG ACWxRXAHsN;[WxO aϕ(n)+1 = aKLOB?

n_Q;jMP;´RPOB?BOaACG

ZnnsdO[_:MX=�KLO8O G:O�sNOjG

n`��WXO A{W�_6OjMTagE:A ACG ACWxRXA)sN;[WxO�UlOgW�I:GQ=�sNO[WzG:O�sNOj_:MP=lKLOgW�sdO[G

n = pqANGZnWXOjG �Qn

q − 1KkI:UlRX=�_:U�O[WH?BA

p i p − 1KkI:UlRX=�_:U�O[WH?BA

q � _6OjMmRT;VGgRXO ϕ(n) = n − 1 − (q − 1) − (p − 1)`

J=�KL_:Ul=l],sC;VGQ?BO,nϕ(n) = (p − 1)(q − 1)

`��e:OjMT;:npq−1 = 1

KLOB?qn i WX= 1 ≤ k < q

ACG[RPOjGQsNACWkq−1 = 1

KLOJ?qn i _6OjMRP;jGgRXO

(kp)(q−1)(p−1) = k(q−1)(p−1)p(q−1)(p−1) = 1KLOB?

qnQU�O�agE:AoagE:=�ANMPAk?BACsN=lM�a[EQA

(kp)ϕ(n) = sq + 1_Q;VMT;L;VU�vjI:G

s`

��E:UlRX=�_:U�=�sC;VGQ?:O\_,O[MkpANGDU�O[Wr?BOgWrU�;[?BO[Wr?Q;

(kp)ϕ(n)+1 = skpq+kp = kpK>OB?

n`V��cB;[s}RT;VKLANGgRXAACU:KL=�WXKLOFRXMT;SRT;VKL=lACGgRXO

?:;>_Q;jMP;1 ≤ k < p

n(kq)ϕ(n)+1 = kq

KLOB?n`J��_6OjMwWXE:_:E:A{W�RPO

0ϕ(n)+1 = 0`

� � ��

�we:O[MP;Qn6WXAC;dsdEQ;jU�agE:=�ANMZGJI:KLANMPOmsNOj_:MP=lKLOmsdOjG

ϕ(n)n i WXAC; e

WXE =�GJYjACMPWXO0ANGZϕ(n)

`6��G[RPOjGQsNACW(Md)e = Mde =

Mkϕ(n)+1_,;VMT;³;jUlv[I:G

k` UQ;VUlRP;´?BACKLO[WxRXMT;VML_,O[MD=lG,?BEQsNsN=lt[G WXOjfQMXA

kagE:AHA{W�RPO)A{W

M`��2O¡;jsC;VfQ;jKLO[W>?BA�?BANKLO[WxRXMT;VM

_Q;VMT;k = 1

`bJ=rWxALsdE:KL_:U�A>_Q;VMT;�;jUlv[I:GkACG[RPOjGQsNACW

M (k+1)ϕ(n)+1 = Mkϕ(n)Mϕ(n)+1 = Mkϕ(n)M = Mkϕ(n)+1 =MKLOJ?

n`

, ;jWX_:EBRPG:= (mWxt[UlO>_:E:A{?BA\MPOjfQ;jMn i d � _Q;VMT;>WX;jf,ACM e RX=�ANG:AoagE:AZp4;[s}RXO[MX=�«C;jM n `��ACMXOLWX=2?BACWPsdE:fQMXAFsdt[K>O^p4;jsdRXO[MX=�«C;jM�ACG�RP=lACK>_6OL_,O[Ul=�G:t[K>=@sdO,nJUlOgW�WXACsNMXANRXOgW�?:AZU�O[W�;jvjANGgRPACWCn i RPEzGJI:KLACMXOL?BAZRP;jM4��ANRP;:n_,ACUl=�vjMT;VGzANGHU@; ��ANf<`

*��<+l�<+ - � 1��C5��C1 � 7 � � " !�5 �� 369 � � 7 " � � 9[+ �GQ;osdO[KL_:EBRP;[?BOjMT;FYJ=lACG:AwA{a[EQ=l_Q;[?:;\?BA�;jMX=lRXKL�NRX=@sN;FKLOB?BE:U�OQnVRP�l_Q= �sN;jK>ACGgRXAjn

232`J��E:A{W�RPOâgE:A

210A{W�;j_:MXO�cB=�KD;j?:;jK>ACGgRXA ¸ �[�j�:n

230A{W�;j_:MXO�cB=�KD;j?:;jK>ACGgRXAKL=lU6KL=lU�U�OjG:A{WNn

109n i 232 × 109

nE:G)GgIQK>ACMXOH?BAD?B=�AN«D?B��vj=lRXOgWNnbsdE,;VGQ?BOmANG)U�O[WFsN�jU�sNE:UlOgWFa[EQAL;[sN;jfQ;VKLO[WZ?BALsdO[GQWx=@?BACMP;jM�G:OgWFANGQsNOjGgRXMT;VKLOgW�sdO[G³GJI:KLACMXOgW?BAkKL=�UlA{WNn�;VIQG�K>=�U�UlO[G:ACWCn6?:A^?B��vj=lRXOgWN`6�<ACG:ANKLO[W�agE:AoRPMP;jfQ;S�x;jMwsdOjG GgIQK>ACMXOgWMPAN_:MPACWXANGgRP;[?BO[W�sdOjKLO�sC;j?BACGQ;jW�?BA^?B��vj=lRXOgWNn_,ACMXODGQO0ACWxRP;jK>OgW�MXA{W�RPMX=�G:v[=�?BOgW;DU@;LfQ;jWXA ¸ �QnQagE:Aoe:ACK>OgW=�GgY[ANGgRP;[?BOD_6OjMTa[EQAFRXANGQANKLO[W ¸ �D?BAC?:O[WCnQWx=�G:O0ACG�E:GQ;LfQ;[WxA

BagE:A�sNOjGJYJ=lACG:A�;oU@;ksNOjKL_:EBRT;j?BO[MP;Qn[agE:A�_:E:AC?:A�WXANM{n[_6OjM�A��ACK>_QUlO,nB = 232

`g��GgRXO[GQsdA{W�MPAN_QMXA{WxACG[RT;VKLO[W�EQG�GJI:KLANMPOksdO[KLOE:GQ;zWxEQsNACWX=lt[G

dk, ..., d0?:OjGQ?BA

0 ≤ di < B_Q;jMP;0sC;j?:;

in6MPAN_:MPACWXANGgRP;jGQ?BODACU�GJI:KLANMPO ∑k

i=0 di × Bi`��OB?BANKLO[W�_,O[G:ANM

�CWxRXOLANGHE:GH;VU�vjO[MX=lRXKLO,hval(d, B) = si(d = ∅) 0

9{1+"�3d0 + B × val(resto(d))?BOjG,?BA

resto([dk , ..., d0])WXA�?:Ad]QG:AFsdO[KLOoU�;^WxE,sdACWX=�tjG

dk, ..., d1`�yuJ=

k = 0ACG[RPOjGQsNACW

resto(d) = ∅ n[U@;kWXEQsdA{Wx=�tjGDYS;[sd�@;:`�|��ANUb=�GJYjANMTWXO>?BAvalA{W

rep(n, B) =9{1

(n = 0) ∅ 9C1!"�3 (c, r) := cr(n, B); r, rep(c, B)`

�agE:�r, rep(c, B)

Wx=�vjGQ= ],sC;0U@;HWxEQsNACWX=lt[GdRP;jU�agE:A

d0 = r i resto(d) = rep(c, B)`�JAL?BANf6ALANGgRXACGQ?BACMoa[EQA

dk, ..., d0?BANGQOVRP;0U�;DKL=@WxKD;0WXEQsNACWX=lt[GHagE:Ad0, ..., dk

n6_,O[MPagE:Ao?:ANG:OjRP;VG�U@;DKL=@WxKD;LpqE:G,sd=�tjGd : {0, ..., k} → {0, ...B − 1} ` ��G�U@;sdO[K>_QEBRP;[?BOjMT;:nJG:OLeQ; i =�«CagE:=�ANMT?:; i ?BACMXA{sTeQ;:`��OB?BACKLO[W�?:ACsd=�M�agE:A

val(rep(n, B), B) = nn,_,ACMXODGQO0G:ACsNACWP;VMP=�;jKLANGgRXA

rep(val(d, B), B) = dn,_,O[MPagE:AoWX=

dRX=�ANG:A

;VU�vjE:GQO[W\�H;HU�;H=�«CagE:=�ANMT?:;Qnb_,O[MoA��ACK>_QUlOdk = dk−1 = 0

n2YS;VG¡;H?BACWP;V_,;VMPACsdACMFANGrep(val(d, B), B)

n<_6OjMTagE:ArepG:O

_:MPOJ?:EQsdAFG:=�G:vjIQGH�L;>U@;^=�«CagE:=�ANMT?:;Q`�we:O[MP;Qn[?:;[?:;jW�?BO[W�sN;[?BANGQ;[W

d, enJ�{sNtjKLOkYS;VKLOgW�;oACG:vjACGQ?BMT;VM�U@;ksC;j?BACGQ;oagE:A�MXAC_:MXA{WxACGgRP;FU@;kWXE:KD;

val(d) + val(e)�

�r;VMPACsdAop4�jsd=�U�MXA{Wx_6OjG,?BANM{hrep(val(d) + val(e))

`��ACMXOzACWXO�G:OmG:OgW�; i EQ?:;0ANG U@;msdOjKL_:E:RP;j?:OjMT;D_,O[MPagE:A^_:MPACWXE:_,O[G:AâgE:AU�;kpqE:GQsN=lt[GvalAdcB=@W�RPA�ACGzU@;^KL�[agE:=lG,;:n i ANGgRXO[GQsdA{W�RP=lACG:A\agE:A�_QMXOB?BEQsN=lM�;VU�vjEQGQ;oMPAN_:MPACWXANGgRT;jsd=�tjGz?BA val(d)

nB_6ANMPOdA{W�U@;

I:G:=@sN;>MPAN_:MPACWXANGgRP;[sd=�tjG�agE:AosNOjG:OBsNANKLO[WC`B�a[EQ�2E:G�;VU�vjO[MX=lRXKLOQhd + e = suma(d, e, 0)� 3 " � �suma(d, e, carreo) =9C1

(d = ∅) 9C1 (carreo = 0) e9C1!"�3 �V�B¤L¥N�Q� �S�

(e)9C1!"�3)9{1(e = ∅) 9{1 (carreo = 0) d

9{1+"�3 �V�B¤L¥N�Q� �S�(d)9C1!"�3

9{1(d0 + e0 + carreo < B) (d0 + e0 + carreo), suma(resto(d), resto(e), 0)9{1+"�3

(d0 + e0 + carreo − B), suma(resto(d), resto(e), 1)� 3 " � ��V�B¤L¥N�Q� �S�

(d) =9C1(d = ∅) [1]9C1!"�39{1

(d0 = B − 1) 0,�S�B¤D¥d�,� �V�

(resto(d))9{1+"�3(d0 + 1), resto(d)J=

B = 2l?BOjG,?BA

lA{WwU@;DU�OjG:v[= RPEQ?�?BA^_Q;VU@;Vf:MT;D?BAÛ�;DKD�[a[EQ=lGQ;QnQACU�_Q;[WxO

d0 + e0 + carreoWxAoe,;jsdA^?B=lMPACsdRP;jK>ACGgRXA

sdO[GzU@;L;VMP= RPK>�NRX=@sN;>=�GQsdO[MX_6OjMT;j?:;kANGHU@;^KD�[a[EQ=lGQ;Qn:agE:Ao?:;>ANUbMPACWXE:UlRP;j?:O>KLOB?BE:U�OB i ;�YJ=@WX;>Wx=beJE:f6OD?BACWXf6OjMT?BAj`��;jMP;LKÊ:UlRX=�_:Ul=@sN;[sd=�tjG2nQeQ; i agE:Aoe,;jsdACME:GQ;0WXE:fBpqEQGQsd=�tjG�agE:A\KÊ:U RP=l_QUl=@sN;D_6OjM�E:G�WxO[UlO0?:�lv[= RPOQ`Q��;LKkE:UlRX=�_:U�=�sC;jsd=�tjG�?:AU�;)sdO[KL_:EBRP;[?BOjMT;�?:;)?BO[f:UlAm_:MPACsN=�WX=lt[G2n�ACWL?BACsN=lML?BOgW>?:�lv[= RPO[W^fQ;jWXA

B`yq��WxRXO¡sdO[MXMPACWX_6OjGQ?BAz; KLANKLO[MX=�«C;jMkU@;�RT;Vf:U@;³?:A

KkE:UlRX=�_:U�=�sC;jsN=lt[GmACGHU�;LA{WXsNE:ANU@;L_:MP=lKD;VMP=@;:`�|�BAksdO[MXMPAd;LU�;L=�«CagE:=�ANMT?:;DsdO[G

0, d`:��G�U�E:vg;VMw?BA\KÊ:U RP=l_QUl=@sN;jMw_,O[M�sN;[?:;LE:G:O

?BAk?B��vj=lRXOgW i WXE:KD;VMrU�O[W k MPACWXE:UlRP;[?BO[WCnVWxEQKL;jKLO[W�sC;j?:;ZMPACWXE:U RT;j?BOo;FU@;\;jsNE:KkEQU�;[sd=�tjGD?BAU�O[W�;jG[RPANMP=lO[MXA{W�ANGDACU:KLOjKLANGgRXOagE:AFACWxRX�\U�=�WxRXO,`� =�YJ=�WX=lt[G�WxA\?BAN]QG:AosN;[Wx=b=�vjE,;VUbagE:A\_Q;VMT;ÛlOgW�GgIQK>ACMXOgW�_:E:MPO[WC`:JAC;jG

d, esC;j?BACGQ;jW�ACGmU@;>fQ;jWXA

Bh

crB(d, e, B) =9C1(d < e) (∅, d)9C1!"�3(c, r) := crB(d, [0, e], B)(c1, r1) := cr(r, e, B)([0, c] + c1, r1)��; i ?BO[W�_:MXO[f:U�ANKD;jW_6ANG,?B=lACGgRXACWZ;jagE:�u`�wG:O0A{W�sNOjKLO�WXAkeQ;[sdA�� d < e

�T`��2O�?:A��x;VKLOgW�sdO[KLODA��ANMTsd=@sd=�OQ`,��U�OVRXMPO0A{WNn�{sdOjKLO0WXAkeQ;[sdA

cr(r, e, B)�0��EQ;VGQ?:O

BANMT; ¸ �D?:= ��=�KLO[WZa[EQAo_,OB?B�@;VKLOgWRXO[UlACMP;jMU�;[Ww_6O[WX=lfQUlA{W R WXEQWxRXMT;jsNsN=lO[G:ACWwANG��CWxRXO,n

� �Q� ��#� ��?��E�?� �� _6ANMPOkG:O>ACW�RP;jG0p4�[sd=�U�;jvjEQ;jGgRP;VM

232 − 1WxEQWxRXMT;jsCsd=�OjG:A{WN`j� U@;>?B=lYJ=@Wx=�tjGm?BAZU�;^KD�jagE:=�GQ;oG:OL; i EQ?:;^;jagE:�6_,O[MPagE:A r i e

WxO[GGJI:KLANMPO[W�vjMT;VGQ?BA{WN`[��ANMPOkWx=�sN;jKkf:=@;VKLOgW�;oU@;ofQ;[WxAZ�\_Q;jMP;

crnJACW�?BA{sd=�MCn[EQWX;jM�ACU

cr2?BA�E:GQO^?BAZUlOgW�Ax��ANMTsd=@sd=�O[W�;jGgRXANMP=�OjMPACWCn

WP;Vf6ANKLO[WagE:AF_Q;jMP;B = 232 e,;Vf:MT�^;>U�O0WxEQK>O ¹Q¸ U�U�;jKD;j?:;[W�MXA{sdE:MTWx=�YS;jW�ANG cr2(r, e)

`�<;D?BACKLO[WxRXMT;jsd=�tjG�pqOjMPKD;VU2?BAoagE:A\A{W�RPO[W;VU�vjO[MX=lRXKLOgW�WXOjGm_6OjU�=lG:t[KL=�sNO[W�ANGHACU2GJI:KLANMPOL?:Ao?B�lv[= RPO[W�A{WKLAC?B=�OLRXA{?B=lOgWX;Qn

_6ANMPO>_6ANG,WX�jGQ?BOjU�OLE:Gm_,OBsdODA{WsdMPAN��f:U�Aj`��; i KkEQsTeQO[Wr_:MPOjfQUlACKL;[Wr?BAwGJI:KLANMPO[W�vjMT;VGQ?BA{WrsdE i ;\WxO[UlE,sd=�tjGL_,O[Ul=�G:t[K>=@sN;\?:AN_6ANGQ?BAZ?BA�agE:AwACUQsN�jU�sNE:U�Oo?BANU,KLsC?0WXAC;_6OjU�=lG:t[KL=�sNOQ`B�rANM/&FGJEBRPe2nBRPOjKLO0�k_,;VMT;ÊQGQ;L?BANKLO[WxRXMT;jsN=lt[Gz?BA\�{W�RPOQ`

��

� �� 1 � 14369 �b+ *��¸ `�JAC;

p_:MP=�K>ODsNOjG:v[MXE:ACGgRXAFsdOjG

3KLtJ?:E:UlO

4n i WXE:_6OjG:vg;>agE:A k

RX=�ANG:AkMP;j�l«FsdEQ;[?BMT;j?:;>ANGZp`'��E:ACWxRXMPAoagE:AFU�;[W

MP;j��sNACW�sdE,;j?BMT;j?:;[W�?:AkACG

ZpWxO[G ±k

p+1

4

`�:` ��OjGQWX=@?BANMPA�ANUB;jU p4;jf,ANRXO { ;:n f2n sVn ?bn�`N`N`Cn Uun Kmn G2n 2n�`N`N`Cn M{n WNn�`N`C`Nn «jn } sdO[G>� P ANU�ANKLANGgRPO[W�yqANUBA{Wx_,;jsd=�O�ACGkf:U@;VG,sdO�sNE:ANGgRT;[|}n;jWX=lv[G:A�;wA{Wx_,;jsd=�OANUgYS;VU�OjM��:n�;w;wACU ¸ n i ;jWX�[WXEQsdA{Wx=�YS;VKLANGgRPA�eQ;jWxRP;w«�sdO[GoYS;VU�OjM��J`V��OjG\A{W�RPOQn�sdEQ;jU�agE:=�ANM2KLACGQWX;V��A?BADRPAdcJRXO WXOjfQMXADA{WxA�;jU p4;jf,ANRXO WXA0_:EQAC?BA0Y[ANMksNOjKLO�E:G¡GJI:KLACMXO�ANcJ_QMXA{WX;[?BO�ANG¡U@;�fQ;[WxAz� P `2��O[MoA��ACK>_QUlO,n�ACUKLANGQWP;S��A�� eQOjU@;��:sdO[MXMPACWX_6OjGQ?BAF;VUbGJI:KLACMXO ¸%P0P ` � R �J`�g| ��;jv[;zE:G´_:MXO[vjMT;VKD;�agE:ALMPACsN=lf6ADE:G´KLANGQWP;S��AL?BADeQ;[W�RT; ¸ � sN;VMT;jsdRXACMXA{W�WXOjf:MPA>ANU�;jU p4;jf,ANRXO i Ojf:RX=�ANG:ADACUGJI:KLANMPO>agE:A\U�A\sdOjMPMPACWX_,O[GQ?BAj`R{|��r;VMT;

p = 71397866609 i q = 71397866609nJEQWX;jGQ?BOmANUrKL�dRPOB?BOm?:A , ;jf:=�G)sNOJ?:= ],agE:A>ANUrKLACGQWX;V��A�� ANU

sdE:MTWXO�� i ?BA{sdOB?B=l],agE:AFANU2KLACGQWX;V��A ¹� � ¸%Rj¹ � R � �!� `²{| ��OjG U�O[WLKL=�WXKLO[Wp = 20511251 i q = 20511319

n�y4ACWxRXAp, qA{W0WXEB],sd=�ANGgRPA�_Q;VMT; ¸ �´sN;jMP;[s}RXACMXA{W>G:O

KD�jWT|�sNOB?B= ]6a[EQA>ACU�KLANGQWP;S��A�� ANU�sNE:MPWXO��ACG)ACU�Wx=@WxRXANKD; , B� sNOjGd = 1009

n i ?BACsNOB?B= ]6a[EQA>ACUrKLANGQWP;S��A� ¹ �S� ¸ P Pj¹0P;R � � `

¨Q| � =@WPsdEBRP=lM\U@;0p4;[s}RP=lf:=�U�=�?:;[?³?:A>KD;jGQ?:;VMFKLANG,WX;V��ACWF?BA>Y�;jUlO[MFKL; i OjMFagE:A pqnbWxE:_6OjGQ=lACGQ?BO�agE:AÂCUrMXA{sdAC_BRXO[M

RX=�ANG:A\ACGmsNE:ANGgRT;Â{WX;>_6O[WX=lfQ=lU�=�?Q;j?b`¹ `��WP;VGQ?:O ��2��nQAdcB_:MPACWP;VM�ACGmf,;jWXA

216 ACU2GJI:KLANMPO 1020 `� `'��WPsdMP=lfQ;HEQG8;VU�vjOjMP=lRXKLO�a[EQA0MXA{sd=�f,Az?BO[WosC;j?BACGQ;jW

c i d?:A�?B��vj=lRXO[WoANG¡fQ;[WxA

Bn i sdO[G[RPACWxRP; 1

WX=�ANU�GJI:KLANMPOMXAC_:MPACWXANGgRP;[?BO�_6OjMoU@;H_:MP=lKLACMP;HsN;[?BANG,;zA{W\KLANG:O[MoagE:ADANU�MPAN_QMXA{WxACG[RT;j?BOm_6OjMoU@;�WxACvjE:G,?:;:n i 0

ANGÓVRPMXO�sN;[WxO,`��MPOjvjMT;VKLAZWxE�;VU�vjO[MX=lRXKLOÂNG ��w��2�w`

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