What is remainder when $5^6 - 3^6$ is divided by $2^3$ (method) I want to know the method through which I can determine the answers of questions like above mentioned one.
PS : The numbers are just for example. There may be the same question for BIG numbers.
Thnx.

We know that 
$x^2-y^2 = (x-y)(x+y) $ and $x^3 + y^3 = (x + y)(x^2 - xy + y^2) $
So,
$5^6 - 3^6 = (5^3)^2 - (3^3)^2 = (5^3 + 3^3) (5^3 - 3^3) $
so, $5^6 - 3^6 = (5 + 3) (5^2 - 3\centerdot 5 - 3^2) (5^3 - 3^3) $
Now, as reminder of $(x\centerdot y $ mod $ y )$ is $0$, dividing above term (with $(5+3)$ as one of the factors) with $2^3(=8)$ will give $0$ reminder.
