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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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This note was uploaded on 02/16/2011 for the course MA 315 taught by Professor Peterturbek during the Spring '11 term at Purdue University Calumet.
 Spring '11
 PeterTurbek

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