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

3
1