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if n is an odd positive integer then show that (n^2 - 1) is divisible by 8. |
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Answer» If 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)⇒ (n^2– 1) is divisible by 8.(n^2– 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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