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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.

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

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

Can you conclude now?

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  • $\begingroup$ ok then $2 ^ n$ will be even and $2 ^ {m - n} - 1$ will be odd?? $\endgroup$ – kartikeykant18 Apr 16 '14 at 11:58
  • $\begingroup$ @kool_kartikey : Yes... so then? $\endgroup$ – user87543 Apr 16 '14 at 11:59
  • $\begingroup$ then n = 3 and m = 6... $\endgroup$ – kartikeykant18 Apr 16 '14 at 12:00
  • $\begingroup$ @kool_kartikey : What else are you looking for? $\endgroup$ – user87543 Apr 16 '14 at 12:07
  • $\begingroup$ math.stackexchange.com/questions/756235/… $\endgroup$ – kartikeykant18 Apr 16 '14 at 12:12
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$$2^m - 2^n = 56\iff 2^n(2^{m-n} - 1) = 2^3\cdot 7 = 2^3( 2^3 - 1) = 56$$

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

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

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