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If n is an odd integer then show that n2-1 is divisible by 8. |
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Answer» Any odd positive integer is in the form of 4p + 1 or 4p+ 3 for some integer p.Let n = 4p+ 1,(n^2 – 1) = (4p + 1) ^2– 1 = 16p^2+ 8p + 1 = 16p^2+ 8p = 8p (2p + 1) (n2– 1) is divisible by 8.(n2– 1) = (4p + 3)^2– 1 = 16p^2+ 24p + 9 – 1 = 16p^2+ 24p + 8 = 8(2p^2+ 3p + 1) n^2 – 1 is divisible by 8. Therefore, n^2 – 1 is divisible by 8 if n is an odd positive integer. |
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