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110ahw1

# 110ahw1 - Math 110A Homework#1 David Wihr Taylor Summer...

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Math 110A Homework #1 David Wihr Taylor Summer 2006 1. Problem 1.1.1: Find the quotient and remainder when a is divided by b : (a) a = 302 , b = 19 (b) a = - 302 , b = 19 (c) a = 0 , b = 19 Answer: (a) 302 = 15 · 19 + 17, so q = 15 and r = 17 (b) - 302 = - 16 · 19 + 2, so q = - 16 and r = 2. Note that q 6 = - 16 and r 6 = - 17 because the division algorithm states that the remainder, r must be less than b and nonnegative. (c) 0 = 0 · 19 + 0, so q = 0 and r = 0. 2. Problem 1.2.1 Find the greates common divisors (a) (56 , 72) =? (b) (24 , 138) =? (c) (143 , 227) =? Answer: (a) 72 = 1 · 56 + 16 56 = 3 · 16 + 8 16 = 2 · 8 + 0 Thus (56 , 72) = 8 by the Euclidean algorithm. (b) 138 = 5 · 24 + 18 24 = 1 · 18 + 6 18 = 3 · 6 + 0 Thus (24 , 138) = 6 by the Euclidean algorithm. 1

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(c) 227 = 1 · 143 + 84 143 = 1 · 84 + 59 84 = 1 · 59 + 23 59 = 2 · 23 + 13 23 = 1 · 13 + 10 13 = 1 · 10 + 3 10 = 3 · 3 + 1 3 = 3 · 1 + 0 Thus (227 , 143) = 1 by the Euclidean algorithm. 3. Problem 1.2.3 Claim. If a | b and b | c , then a | c Proof. Since a | b , there exists q Z such that b = q · a . Similarly, since b | c , there exists r Z such that c = r · b Thus c = r · ( q · a ) = ( r · q ) · a (by associativity) so a | c .
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