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homework1ans - n is even or n is odd If n is even then...

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MA 315, Spring 2011 Homework 1 Answers Due Wednesday, January 26, 2011 Prove the following two statements. Your paper must be typed. Each proof must be written in complete sentences with correct spelling, punctuation and grammar. (1) Let a , b , and c be integers. Prove that if a | b and a | ( b + c ) then a | c . Proof: Let a , b , and c be integers and assume a | b and a | ( b + c ). This means that there exist integers k and r such that ak = b and ar = b + c . Substituting the value of b from the first equation into the second equation yields that ar = ak + c . This yields that ar - ak = c , so a ( r - k ) = c . Note that, since r and k are integers, r - k is an integer. This means that a | c , since there is an integer, namely r - k , that can be multiplied by a to yield c . (2) Let n be an integer. Prove that if n 2 is not divisible by 4, then n 2 is not divisible by 2. Proof: Let n be an integer and assume n 2 is not divisible by 4. We must show that n 2 is not divisible by 2. There are two cases two consider, either
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Unformatted text preview: n is even or n is odd. If n is even, then there exists an integer k such that n = 2 k , therefore n 2 = (2 k ) 2 = 4 k 2 , therefore 4 | n 2 , so n 2 is divisible by 4. But our assumption is that n 2 is not divisible by 4, therefore n cannot be even. So n is odd, which means that there exists an integer k such that n = 2 k + 1, so n 2 = (2 k + 1) 2 = 4 k 2 + 4 k + 1 = 4( k 2 + k ) + 1, which shows that n 2 is not divisible by 4. We can rewrite the lase equation as n 2 = 2(2 k 2 + 2 k ) + 1, which shows that n 2 is odd, since n 2 is 1 more than twice the integer 2 k 2 + 2 k . We have shown that if n 2 is not divisible by 4, then n must be odd, and therefore that n 2 is odd, so n 2 is not divisible by 2. 1...
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