hw3sol - WEEK 3 HOMEWORK SOLUTIONS 3.4-9(a q 2 r 5(b q-11 r...

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Unformatted text preview: WEEK 3 HOMEWORK SOLUTIONS 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 : If a mod m = b mod m , then a = qm + r and b = q prime m + r for integers q, q prime , r by applying the division algorithm. Thus a- b = ( q- q prime ) m , a multiple of m . Hence a ≡ b (mod m ). 3.4-12 : If a ≡ b (mod m ), then we have a- b = dm , for some integer d . Apply the division algorithm to b , writing b = qm + r , with 0 ≤ r < m . Thus b mod m = r . Then we have a = dm + b = ( d + q ) m + r , and since 0 ≤ r < m , we see that a mod m = r as well, completing the proof. 3.4-17 : (a) 1 (b) 2 (c) 3 (d) 9 3.4-21 : By assumption we can write a- b = dm , for some integer d . In addition, since n divides m , we can write m = kn , for some integer k . Then a- b = dkn , showing that a- b is a multiple of n and hence that a ≡ b (mod n )....
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This note was uploaded on 04/05/2010 for the course MATH 55 taught by Professor Strain during the Spring '08 term at Berkeley.

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hw3sol - WEEK 3 HOMEWORK SOLUTIONS 3.4-9(a q 2 r 5(b q-11 r...

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