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We need to show that 4 divides 1-n2 whenever n is an odd positive integer.

If n is an odd positive integer then by definition
n = 2k+1 for some non negative integer, k.
Now 1 - n2 = 1 - (2k+1)2 = -4k2-4k = 4 (-k2-4k).
k is a nonnegative integer, hence -k2-4k is an integer. Thus by definition
of divisibility we conclude that 4 divides 1-n2.

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