homework3ans - q 3 + 27 q 2 + 6 q = 3(9 q 3 + 9 q 2 + 2 q...

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MA 315, Spring 2011 Homework 3 Answers Due Wednesday, February 9, 2011 For this homework assignment you are allowed to use the following fact about in- tegers: If n is an in integer, then n leaves a remainder of either 0, 1, or 2 when n is divided by 3. For example, 6 is divisible by 3, 7 leaves a remainder of 1 upon division by 3, and 8 leaves a remainder of 2 upon division by 3. Your paper must be typed. Each proof must be written in complete sentences with correct spelling, punctuation and grammar. (1) Prove the following: If n is any integer, then n 3 - n is divisible by 3. Proof: Assumee n is an integer. From the division algorithm, n = 3 q + r for some integers q and r , where 0 r < 3. This leads us to consider three cases. (a) If r = 0, then n = 3 q , so n 3 - n = (3 q ) 3 - 3 q = 27 q 3 - 3 q = 3(9 q 3 - q ). Since 9 q 3 - q is an integer, 3(9 q 3 - q ) is divisible by 3, so n 3 - n is divisible by 3. (b) If r = 1, then n = 3 q + 1. Therefore n 3 - n = (3 q + 1) 3 - (3 q + 1) = 27
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Unformatted text preview: q 3 + 27 q 2 + 6 q = 3(9 q 3 + 9 q 2 + 2 q ). Since 9 q 3 + 9 q 2 + 2 q is an integer, 3(9 q 3 + 9 q 2 + 2 q ) is divisible by 3, so n 3-n is divisible by 3. (c) If r = 2, then n = 3 q + 2. Therefore n 3-n = (3 q + 2) 3-(3 q + 2) = 27 q 3 + 54 q 2 + 33 q + 6 = 3(9 q 3 + 18 q 2 + 11 q + 6). Since 9 q 3 + 18 q 2 + 11 q + 6 is an integer, 3(9 q 3 + 18 q 2 + 11 q + 6) is divisible by 3, so n 3-n is divisible by 3 Since n must belong to one of the three cases above, we have proved that n 3-3 is divisible by 3. (2) Prove or disprove: If a , b and c are integers and a | b and b | c , then ab | c . The statement is false. A counterexmple is the following. Let a = 4 and b = 8 and c = 24. Then a | b because 4 | 8 and b | c , since 8 | 24. However it is not true that ( ab ) | c because 32 6 | 24. 1...
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