Posted by
**Don(please check my math)** on
.

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.