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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 3n is divisible by 3. (c) If r = 2, then n = 3 q + 2. Therefore n 3n = (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 3n is divisible by 3 Since n must belong to one of the three cases above, we have proved that n 33 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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 Spring '11
 PeterTurbek
 Remainder, Integers

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