Question

If a and b are odd positive integers and a - b is divisible by 2^n, where n is a positive integer, then
a-b is divisible by
​

Answer 1

Factorizing, we get

a4−b4=(a2+b2)(a2−b2)

a4−b4=(a2+b2)(a−b)(a+b)

Now if a and b are odd positive integers then a+b as well as a−b will be even.

So 2|(a+b) and 2|(a−b).

Also a2 is odd, b2 is odd, hence 2|(a2+b2).

Hence a4−b4 is always divisible by 2,4,8 for any odd positive a and b.

Hope it helps!