Question

What is the remainder if $7^{25}$ is divided by 4 ?

Answer 1

$7^{25}$ can be written $(8-1{)}^{25}$
There are 26 terms in all and first 25 terms are divisible by 8 hence
also by 4 The last term is
$(-1{)}^{25}$
Hence $(8-1{)}^{25}$ can be written 8X - 1 or 4Y - 1
(where Y = 2X) So 4Y - 1 divided by 4 leaves the remainder 3