hw3solsp09

hw3solsp09 - Math 55: Discrete Mathematics UC Berkeley,...

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Unformatted text preview: Math 55: Discrete Mathematics UC Berkeley, Spring 2009 Solutions to Homework # 3 (due February 9) § 3.4, #9: a) q = 2 ,r = 5 b) q =- 11 ,r = 10 c) q = 34 ,r = 7 d) q = 77 ,r = 0 e) q = 0 ,r = 0 f) q = 0 ,r = 3 g) q =- 1 ,r = 2 h) q = 4 ,r = 0 § 3.4, #11: Say a mod m = b mod m . Now by the Division Algorithm and the definition of a mod m , there exist integers q and q such that a = qm + ( a mod m ) and b = q m + ( b mod m ). Then a- b = qm + ( a mod m )- q m- ( b mod m ) = ( q- q ) m. By Theorem 1(ii), m | ( q- q ) m ; thus m | a- b , and this is the definition of the conclusion a ≡ b (mod m ). § 3.4, #12* Say a ≡ b (mod m ). The definition of this is that m | a- b . As in #11, by the Division Algorithm and the definition of a mod m , there exist in- tegers q and q such that a = qm +( a mod m ) and b = q m +( b mod m ). Then a- b = ( q- q ) m + ( a mod m )- ( b mod m ) . By Theorem 1(ii), m | ( q- q ) m ; we have m | a- b , so by Theorem 1(i), m | ( a mod m )- ( b mod m ). By definition, there exists some)....
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This note was uploaded on 10/10/2009 for the course MATH 55 taught by Professor Strain during the Spring '08 term at Berkeley.

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hw3solsp09 - Math 55: Discrete Mathematics UC Berkeley,...

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