Factor of a Mersenne number Why is it true that if 7 divides 91 then $(2^7-1) $ divides $(2^{91}-1)$?
1) $2^{91}-1$
$7|91  \implies (2^7-1)|(2^{91}-1)$
$\implies 2^7-1$ is factor 
2) $2^{1001}-1$
$7|1001  \implies (2^7-1)|(2^{1001}-1)$
$\implies 2^7-1$ is factor 

We have $$2^{91}-1 = (2^7)^{13}-1 = (2^7-1)((2^7)^{12}+...+2^7+1)$$
More generally:
$$2^{ab}-1 = (2^a)^{b}-1 = (2^a-1)((2^a)^{b-1}+...+2^a+1)$$
so $2^a-1\mid 2^{ab}-1$
And we can replace $2$ with any other number. 
