Question

If n is an odd integer theb show that n^2-1 is divisible by 8

Answer 1

Any odd integer is of the form :-

4q + 1 or 4q + 3, where q is some integer

Since, n is an odd integer

Therefore,

First Case : If n is of the form 4q + 1 => n² - 1 = (4q + 1)² - 1

=> n² - 1 = 16q² + 8q + 1 - 1

=> n² - 1 = 8q(2q + 1)

Clearly, it is divisible by 8

Second Case : If n is of the form 4q + 3

=> n² - 1 = (4q + 3)² - 1

=> n² - 1 = 16q² + 24q + 9 - 1

=> n² - 1 = 8(8q² + 3q + 1)

Clearly, it is also divisible by 8

Thus, we can conclude that, n² - 1, where n is an odd integer, is divisible by 8.

Hope, it'll help you.....