Spot the Difference

Subtract the two numbers to obtain $2^{2016}-2^{2010}$ . The GCD should divide this difference. Factor out $2^{2010}$ because any power of 2 greater than 1 will not divide either number. This leaves 63, and since any power of $2^{6}$ leaves a remainder of 1 when divided by 63, both $2^{2016}-1$ and $2^{2010}-1$ are divisible by 63.