If n is an odd integer, then show that n 2 – 1 is divisible by 8.

Question

Philoid · Accepted Answer

Let a be of the form n²– 1, where n is odd

i.e. n = 2k + 1, where k is an integer.

⇒ a = (2k + 1)² -1

⇒ a = 4k² + 1 + 4k-1

⇒ a = 4k² + 4k = 4k(k + 1)

At k = 1, a = 4(1)(1 + 1) = 4 × 2 = 8, which is divisible by 8.

At k = 2, a = 4(2)(1 + 1) = 8 × 2 = 16 which is divisible by 8.

And so on.

Hence, we can conclude from above two cases that if n is odd, then n² -1 is divisible by 8.