Find $m$ and $n$

Question

Two finite sets have m and n elements. Thew total number of subsets of the first set is 56 more than the two total number of subsets of the second set. Find the value of $m$ and $n$.

The equation to this question will be $2 ^ m$ - $2 ^ n = 56$.

But I don't know how to solve this equation.

Answer 1

$2^m-2^n=56\Rightarrow 2^n(2^{m-n}-1)=56$

Look for factorization of $56$ into two numbers $a,b$ such that $a$ is even and $b$ is odd..

Can you conclude now?

Answer 2

Hint: min(n,m) is 8 as denominator of 56

Answer 3

$$2^m - 2^n = 56\iff 2^n(2^{m-n} - 1) = 2^3\cdot 7 = 2^3( 2^3 - 1) = 56$$

Answer 4

Let $k = m-3$ and $l=n-3$, then $$ 2^k-2^l = 56/2^3 = 7. $$ Now determine the values of $k$ and $l$.