n² -1 is divisible by 8, if n is

Let a be of the form n²– 1, n can be odd or even.

Let us consider two cases.

Case I If n is even i.e. n = 2k, where k is an integer.

⇒ a = (2k)² -1

⇒ a = 4k² -1

At k = 1, = 4 (1)² -1 = 4 -1 = 3, which is not divisible by 8.

At k = 2, a = 4 (2)² -1 = 16-1 = 15 which is not divisible by 8.

Case II If 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.

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