Any odd positive integer is of the form 4m+1 or 4m+3 for some integer m
for n = 4m+1
n^2-1 = (4m+1)^2 - 1
= 16m^2 + 8m
= 8(2m^2 + m)
hence it is divisible by 8
for n = 4m+3
n^2 - 1 = 16m^2 + 24m + 8
= 8(2m^2 + 3m + 1)
it is also divisible by 8