Question

if n is an odd integer show that n²-1 is divisible by 8

Answer 1

Since n is odd n=4m+1n=4m+1 or n=4m+3n=4m+3.

In the first case n2−1=(n−1)(n+1)=4m⋅(4m+2)=8m(2m+1)n2−1=(n−1)(n+1)=4m⋅(4m+2)=8m(2m+1), while in the second case n2−1=(n−1)(n+1)=(4m+2)⋅(4m+4)=8(2m+1)(m+1)n2−1=(n−1)(n+1)=(4m+2)⋅(4m+4)=8(2m+1)(m+1).

So n2−1n2−1 is divisible by 8 if nn is odd.