Question

If n is an odd integer, then show that n2 - 1
is divisible by 8.​

Answer 1

Step-by-step explanation:

n is an odd integer. any odd integer can be expressed as 2a-1 or 2a+1 where a is any positive integer

let n = 2a+1

n^2 - 1

= (2a+1)^2 - 1

= (2a+1-1)(2a+1+1)

= (2a+2)×(2a)

= 4a(a+1)

a(a+1) is the product of any two consecutive numbers a and a+1. We know that product of any two consecutive numbers is always even. so a(a+1) can be expressed as 2k.

Now,

n^2 - 1

= 4a(a+1)

= 4×2k = 8k and this is divisible by 8

Hence, If n is an odd integer, then n^2 - 1 is divisible by 8.