How to find the values of m & n in $2^{m} - 2^{n} = 56 ?$

Question

I've tried this in two ways :

First Way :

$2^{m}- 2^{n} = 56$

$2^{m}(1-2^{n-m})= 2^{3} × 7$

Comparing the exponents, we get $m = 3$

And,

$1-2^{n-m} = 7$

$n = 6$

Second Way :

$2^{m}- 2^{n} = 56$

$2^{n}(1-2^{m-n})= 2^{3} × 7$

Comparing the exponents, we get $n = 3$

And,

$1-2^{m-n} = 7$

$m = 6$

Is what I've done correct ? If this method is correct, why I'm getting different answers, just by factorising it in two different ways ?

Answer 1

It can only be a solution because $2^m-2^n=56>0\Rightarrow2^m-2^n>0$, so $m>n$, and therefore you can only do: \begin{align} 2^n(2^{m-n}-1)=56=2^3\times7 \end{align} Where do you have to $n=3$ y $m=6$.