If n is an odd integer, then show that n2 – 1 is divisible by 8.


Let a be of the form n2– 1, where n is odd


i.e. n = 2k + 1, where k is an integer.


a = (2k + 1)2 -1


a = 4k2 + 1 + 4k-1


a = 4k2 + 4k = 4k(k + 1)


At k = 1, a = 4(1)(1 + 1) = 4 × 2 = 8, which is divisible by 8.


At k = 2, a = 4(2)(1 + 1) = 8 × 2 = 16 which is divisible by 8.


And so on.


Hence, we can conclude from above two cases that if n is odd, then n2 -1 is divisible by 8.


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