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MIT6_042JS10_lec22

MIT6_042JS10_lec22 - Congruence mod n Mathematics for...

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4/2/08 2:20PM ‹#› Albert R Meyer, March 31, 2010 Congruences: arithmetic (mod n) Mathematics for Computer Science MIT 6.042J/18.062J 8W.1 Albert R Meyer, March 31, 2010 Congruence mod n Def : a b (mod n) iff n|(a - b) example: 30 12 (mod 9 ) since 9 divides 30 - 12 8W.3 Albert R Meyer, March 31, 2010 example: 66666663 788253 (mod 10) WHY? 66666663 - 788253 xxxxxxx 0 Congruence mod n 8W.4 Albert R Meyer, March 31, 2010 a b (mod n) iff rem(a,n) = rem(b,n) Remainder Lemma example: 30 12 (mod 9) since rem( 30 , 9 ) = 3 = rem( 12 , 9 ) 8W.8 Albert R Meyer, March 31, 2010 a b (mod n) iff rem(a,n) = rem(b,n) r b,n abbreviate: 8W.9 Remainder Lemma Albert R Meyer, March 31, 2010 a = q a n + r a,n b = q b n + r b,n if rem’s are =, then a-b=(q a -q b )n proof: (if) so n|(a-b) (only if) proof similar 8W.11
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4/2/08 2:20PM ‹#› Albert R Meyer, March 31, 2010 a b (mod n) iff rem(a,n) = rem(b,n) QED 8W.14 Remainder Lemma Albert R Meyer, March 31, 2010 Corollary: a rem(a,n) (mod n) pf: 0 r a,n < n , so rem(r a,n ,n) = r a,n Remainder arithmetic 8W.15 Albert R Meyer, March 31, 2010 symmetric a b (mod n) implies b a (mod n) transitive a b & b c (mod n) implies a c (mod n) More Corollaries 8W.17
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