Question

if n is an odd intiger show that n^2-1 is divisible by 8​

Answer 1

We know every odd integers leave remainder 1 on division by 2.

So, let n = 2k + 1, for every integer k.

Now,

n² - 1

=> (2k + 1)² - 1

=> 4k² + 4k + 1 - 1

=> 4k² + 4k

=> 4 · k(k + 1)

Here we seems 4 is multiplied to the product of two consecutive integers.

This product of two consecutive integers is always even. Let me show you.

If k is odd, then k + 1 will be even. So k(k + 1) will also be even.

If k is even, there will be no matter what k(k + 1) will be. It'll also be even.

Hence let k(k + 1) = 2m, for every whole number m. So,

4 · k(k + 1)

=> 4 · 2m

=> 8m

Here n² - 1 seems as a multiple of 8.

Hence Proved!