# sol1 - Math 465 Solution Set 1 1. Let a, b, c Z. Prove each...

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Math 465 Solution Set 1 Ae Ja Yee 1. Let a, b, c Z . Prove each of the following. (a) If ac | bc and c 6 = 0, then a | b . (b) If a | b , then ac | bc . Solution. (a) Since bc = acx for some x Z and c 6 = 0, b = ax . (e) b = ax for some x Z , so bc = acx . 2 2. Prove that if n is odd then 8 | n 2 - 1. Solution. Since n is odd, n = 2 k + 1 for some k Z . So, n 2 - 1 = (2 k + 1) 2 - 1 = 4 k ( k + 1) . Since k and k + 1 are consecutive numbers, one of them must be even. Therefore, 4 k ( k + 1) is divisible by 8. 2 3. Prove that 5 | n 5 - n . Solution. Method 1: n 5 - n = n ( n 4 - 1) = n ( n 2 + 1)( n + 1)( n - 1) If 5 | n , then clearly 5 | n ( n 2 + 1)( n + 1)( n - 1). If 5 | n + 1 or 5 | n - 1, then 5 | n ( n 2 + 1)( n + 1)( n - 1). Otherwise, n has remainder 2 or 3 when divided by 5. In either case, n 2 + 1 is a multiple of 5, namely n 2 + 1 = (5 k + 2) 2 + 1 = 25 k 2 + 20 k + 5 = 5(5 k 2 + 4 k + 1) for n = 5 k + 2 n 2 + 1 = (5 k + 3) 2 + 1 = 25 k 2 + 30 k + 10 = 5(5 k 2 + 6 k + 2)

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## This note was uploaded on 07/23/2008 for the course MATH 465 taught by Professor Yee during the Spring '08 term at Penn State.

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sol1 - Math 465 Solution Set 1 1. Let a, b, c Z. Prove each...

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