Question

Can you find two integers $m,n$ such that $2^{m+n}=2^{mn}$?

Answer 1

Equating the powers of $2^{m+n}=2^{mn}$ that is $m+n=mn$.

Now we have to find such integers so that the sum of the integers is equal to their product and those integers are $0$ and $2$:

Let us first take $m=2$ and $n=2$, we get

$2+2=2\times 2$

$\Rightarrow 4=4$

which is true

Now take $m=0$ and $n=0$, we get

$0+0=0\times 0$

$\Rightarrow 0=0$

which is true

Hence the integers are $m=0,2$ and $n=0,2$.