<s>[INST] 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 
 [/INST] It may be illustrative to write the numbers out in binary.  I'll use $2^{21} - 1 = (2^7)^3 - 1$ instead of $2^{91} - 1$, since it's shorter:
$$\begin{aligned}
2^{21} - 1
&= \underbrace{111111111111111111111}_{21\text{ digits}}\,\vphantom1_2 \\
&= \underbrace{1111111}_{7\text{ digits}}\,\underbrace{1111111}_{7\text{ digits}}\,\underbrace{1111111}_{7\text{ digits}}\,\vphantom1_2 \\
&= 1111111_2 \times 100000010000001_2 \\
&= (2^7 - 1) \times (2^{14} + 2^7 + 1).
\end{aligned}$$
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