Question

if n is an odd positive integer,
then prove that
n^2 -1 is divisible by 8

Answer 1

if n is odd, n=2m+1 for some m
n^2-1
= (2m+1)^2-1
= 4m^2 + 4m + 1 - 1
= 4m(m+1)

Now, either m or m+1 is even, so m(m+1) is a multiple of 2

So, 4m(m+1) is a multiple of 8