I stumbled upon a solution to the so-called "The Math Problem of the Month" for October 2018. 
<https://www.aus.edu/sites/default/files/PastProblems_October18.pdf>

Now, I have to admit I am very confused by the last few steps. 

It says there: 

"If $n$ is a multiple of $3$, then $2^{7n}-1$ has the factor $2^{3}-1$" 

How do you get the factor out of there? If you set $n=3$ for example,you get $2^{7\*3}-1$, but you can't factor out $2^{3}-1$, so what is the method behind that argument? 

"When $n$ is not a multiple of $3$, it is easy to see that $7$ does not divide $2^{7n}-1$" 

Why? Because you can't extract the factor which you couldn't do in the first place? 

"A similar argument shows that 

There are $2\*953$ values for $m= 1, 2$"

First, shouldn't it be $m=1,3$? 

Second, shouldn't it be $2\*476$ or am I completely missing something here? 

I appreciate any help.