Given
if n is odd then 8 divides n2-1
Proof
We can proof the given statement directly.
Let n ∈ O
Let,
P = n2 – 1
We need to prove that P is divisible by 8
P = n2 – 12
P = (n + 1)(n – 1)
We know that n is odd. This implies that n + 1, n – 1 must be even. Let n – 1 = x
P = x * (x + 2)
Both x and x+2 are even. When we divide x by 2, we have 2 cases
CASE 1: x / 2 = y, y is odd
Divide P by 8
CASE 2: x / 2 = y, y is even
In both the cases, P is completely divisible by 8.
Hence proved.