sol_ch6_congruences

# sol_ch6_congruences - .kdswhu 9.rqjuxhqfhv 914...

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Fkdswhu 9 Frqjuxhqfhv 914 Frqjuxhqfhv 41 +d, 77 46 @ 64 lv qrw glylvleoh e| 6 = Khqfh/ 77 9 46+prg 6, = Wkhuhiruh/ wkh vwdwhphqw lv idovh1 +e, 458 58 @ 433 lv glylvleoh e| 7 = Khqfh/ 458 58+prg 7, = Wkhuhiruh/ wkh vwdwhphqw lv wuxh1 +f, :8 65 @ 76 lv qrw glylvleoh e| 8 = Khqfh/ :8 9 65+prg 8, = Wkhuhiruh/ wkh vwdwhphqw lv idovh1 +g, 88 8@ 93 lv glylvleoh e| 7 = Khqfh/ 88 8+prg 7, = Wkhuhiruh/ wkh vwdwhphqw lv wuxh1 +h, ; 7 [email protected]<6 lv qrw glylvleoh e| 44 = Khqfh/ ; 7 9 6+prg 44, = Wkhuhiruh/ wkh vwdwhphqw lv idovh1 +i, : 5 4+prg 43, lpsolhv wkdw : 7 4+prg 43, = Wkhuhiruh/ wkh vwdwhphqw lv wuxh1 61 Zh kdyh wr qg dq lqwhjhu u vxfk wkdw 3 u? 4; dqg 4\$.5\$.6\$. .<<\$.433\$ u +prg 4;, = Qrz 4\$ 4+prg 53, 5\$ 5+prg 53, 6\$ 9+prg 53, 7\$ 7+prg 53, 8\$ @ 8 7 6 5 3+prg 53, 9\$ @ 9 8 7 6 5 3+prg 53, q \$ 3+prg 53, iru doo q 8 = Wkxv/ 4\$ . 5\$ . 6\$ . +4.5.9.7,+prg53, 46+prg 53, = Khqfh/ 46+prg 53, = Vr wkh uhpdlqghu lv 46 = 81 E| wkh glylvlrq dojrulwkp/ Wkhruhp 51419/ zh nqrz wkdw wkhuh h{lvw txrwlhqw t dqg wkh uhpdlqghu u vxfk wkdw : 63 @ t 7. u> 3 7 = Wklv phdqv wkdw wkh uhpdlqghu u vdwlv hv wkh frqjuxhqfh : 63 u +prg 7, > zkhuh 3 7 = Qrz : 4+prg 7, , : 5 + 4, 5 +prg 7, e| Wkhruhp 914149+lll, , : 5 4+prg 7, , : 5 48 4 48 +prg 7, e| Wkhruhp 914149+lll, , : 63 4+prg 7, 1 Khqfh/ wkh uhpdlqghu lv 4 = :1 Ohw { eh wkh uhtxluhg qxpehu1 Wkhq { ;+prg 43, , { .5 43+prg 43, , { 3+prg 43, = { :+prg <, , { <+prg <, , { 3+prg <, = :6

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:7 91 Frqjuxhqfhv Vlploduo|/ { .5 3+prg ;, >{ 3+prg :, > dqg { 3+prg 9, = Wkxv/ { lv wkh ohdvw frpprq pxowlsoh ri 43 > < > ; > : dqg 9 = Qrz OFP ri 43 > < > ; > : > dqg 9 lv 5853 = Khqfh/ { . 5 @ 5853 = Wklv lpsolhv wkdw { @5853 5 @ 584; = <1 Ehfdxvh d e +prg p , dqg f g +prg p , >p glylghv d e dqg f g= Khqfh/ wkhuh h{lvw lqwhjhuv n dqg w vxfk wkdw d e @ pn dqg f g @ pw= Wkhq + d e , { .+ f g , | @ pn{ . pw| @ p + n{ . w| , = Wklv vkrzv wkdw + d{ . f| , + e{ . g| ,@ p + n{ . w| , = Wklv lpsolhv wkdw p glylghv + d{ . f| , + e{ . g| , = Khqfh/ d{ . f| + e{ . g| ,+prg p , = 441 +d, Ehfdxvh 5 0 : > 49 grhv qrw glylgh 8<334537: = +e, Ehfdxvh 5 7 0 <545 > 49 grhv qrw glylgh 4:<<545 = +f, Ehfdxvh 5 7 m 5349 > 49 glylghv 7<35349 = 461 +d, [email protected] 3+prg 6, dqg 9 3+prg <, = Khqfh/ 6 glylghv 8745684 exw < grhv qrw glylgh 8745684 = +e, <.3.<.6.<.6.< 3+prg 6, dqg 9 3+prg <, = Khqfh/ 6 glylghv <3<6<6< exw < grhv qrw glylgh <3<6<6< = +f, +43353434, 6 @4 6 : .3 6 9 6 8 6 7 6 6 .4 6 5 6 4 6 3 6 : 6 7 6 5 = Wklv lpsolhv wkdw 6 grhv qrw glylgh +43353434, 6 = Iurp wklv lw dovr iroorzv wkdw < grhv qrw glylgh +43353434, 6 = +g, 7.:.9.3.4.;.[email protected]: 3+prg <, = Khqfh/ 6 glylghv 7:934;4 dqg < glylghv 7:934;4 = 481 +d, 8363955 lv glylvleoh e| 99 li dqg rqo| li 8363955 lv glylvleoh e| 5 > 6 > dqg 44 = Ehfdxvh 8363955 lv dq hyhq qxpehu/ lw lv glylvleoh e| 5 = Qrz wkh vxp ri wkh gljlwv ri 8363955 lv 8.3.6.3.9.5.5 @ 4; > zklfk lv glylvleoh e| 6 = Khqfh/ 8363955 lv glylvleoh e| 6 = Djdlq 8363955 lv glylvleoh e| 44 li dqg rqo| li 5 5.9 3.6 3.8 lv glylvleoh e| 44 = Exw 5 [email protected] lv qrw glylvleoh e| 44 = Khqfh/ 8363955 lv qrw glylvleoh e| 99 = +e, 836<5369 lv glylvleoh e| 99 li dqg rqo| li 836<5369 lv glylvleoh e| 5 > 6 > dqg 44 = Ehfdxvh 836<5369 lv dq hyhq qxpehu/ lw lv glylvleoh e| 5 = Qrz wkh vxp ri wkh gljlwv ri 836<5369 lv 8.3.6.<.5. [email protected]; > zklfk lv qrw glylvleoh e| 6 = Wkxv/ 836<5369 lv qrw glylvleoh e| 6 = Khqfh/ 836<5369 lv qrw glylvleoh e| 99 = +f, 6363657 lv glylvleoh e| 99 li dqg rqo| li 6363657 lv glylvleoh e| 5 > 6 > dqg 44 = Ehfdxvh 6363657 lv dq hyhq qxpehu/ lw lv glylvleoh e| 5 = Qrz wkh vxp ri wkh gljlwv ri 6363657 lv 6.3.6.3.6.5.7 @ 48 > zklfk lv glylvleoh e| 6 = Khqfh/ 6363657 lv glylvleoh e| 6 = Djdlq 6363657 lv glylvleoh e| 44 li dqg rqo| li 7 5.6 lv glylvleoh e| 44 = Qrz 7 [email protected] lv glylvleoh 44 = Wkxv/ 6363657 lv glylvleoh e| 44 = Khqfh/ lw iroorzv wkdw 6363657 lv glylvleoh e| 99 = 4:1 Qrz q @8 49 9 .6 49 8 . I 49 7 49 6 .8 49 5 . D 49 4 .9 = Qrz 49 4+prg 4:, > 49 5 4+prg 4:, > 49 6 4+prg 4:, > 49 7 4+prg 4:, > 49 8 4+prg 4:, > 49 9
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## This note was uploaded on 11/11/2009 for the course CSE 2353 taught by Professor Bhanukapoor during the Spring '09 term at SMU.

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sol_ch6_congruences - .kdswhu 9.rqjuxhqfhv 914...

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