# Find $m$ and $n$

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.

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

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

Can you conclude now?

• ok then $2 ^ n$ will be even and $2 ^ {m - n} - 1$ will be odd?? – kartikeykant18 Apr 16 '14 at 11:58
• @kool_kartikey : Yes... so then? – user87543 Apr 16 '14 at 11:59
• then n = 3 and m = 6... – kartikeykant18 Apr 16 '14 at 12:00
• @kool_kartikey : What else are you looking for? – user87543 Apr 16 '14 at 12:07
• math.stackexchange.com/questions/756235/… – kartikeykant18 Apr 16 '14 at 12:12

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

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

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