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Sorry new to this forum and don't know how to format:

If $m,n\in\Bbb N$ satisfy $6^{2m+2}\cdot 3^n=4^n\cdot 9^{m+3}$, then $n$ and $m$ must be ... what?

This is for my discrete mathematics class. I tried taking ln/log of both sides but got nowhere. Primes didn't go anywhere for me either.

Edit: Maybe I have to use Fundamental Theorem of Arithmetic?

Brian M. Scott
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user1010282
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    Your guess is correct, write each side as $2^a3^b$. – vadim123 Dec 14 '13 at 01:59
  • This is what I was having trouble with, I couldn't get anywhere with it.. I rewrote it as 2^2m+2(3^2m+2) etc.. – user1010282 Dec 14 '13 at 02:01
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    You have $2^{2m+2}\cdot3^{2m+2}\cdot3^n=2^{2n}\cdot3^{2(m+3)}$, or $2^{2m+2}\cdot3^{2m+2+n}=2^{2n}\cdot3^{2m+6}$. This gives you two equations in the unknowns $m$ and $n$. – Brian M. Scott Dec 14 '13 at 02:12

3 Answers3

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Hint $\rm\displaystyle\ \ \frac{\color{#c00}{3^n}}{\color{#0a0}{4^n}} = \frac{9^{m+3}}{(3\cdot 2)^{2(m+1)}} = \frac{9^{m+3}}{9^{m+1} 4^{m+1}} = \frac{\color{#c00}{9^2}}{\color{#0a0}{4^{m+1}}}\ \Rightarrow\ \color{#c00}{n = \ldots}\ \Rightarrow\ \color{#0a0}{m = n-1 = \ldots}$

Remark $\ $ This implicitly uses the uniqueness of reduced fractions, which is equivalent to uniqueness of prime factorizations (used in the other answers/hints).

Bill Dubuque
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Hint: $$\begin{align*}6^{2m+2}\cdot 3^n&=4^n\cdot 9^{m+3}\\ \\ \longrightarrow \color{red}{2^{2m+2}}\cdot\color{blue}{3^{2m+2+n}}&=\color{red}{2^{2n}}\cdot \color{blue}{3^{2m+6}} \end{align*}$$ Two integers are the same iff they have the same prime factors. Thus you can equate the exponents corresponding to each prime and solve for the resulting system of equations.

E.O.
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We have the following: $$ 6^{2m+2}\cdot 3^n=(2\cdot 3)^{2m+2}\cdot 3^n = 2^{2m+2}\cdot 3^{2m+2}\cdot 3^n = 2^{2m+2}\cdot 3^{2m+2+n} $$ And $$ 4^n\cdot 9^{m+3} = 2^{2n} \cdot 3^{2m+6}. $$ Equating them gives: $2^{2m+2}\cdot 3^{2m+2+n} = 2^{2n} \cdot 3^{2m+6}.$. So we need to find $m,n \in \mathbb{N}$ such that $2m+2=2n$ and $2m+2+n=2m+6$, because prime factorization is unique. The rest I will leave up to you.

user112167
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