Gmat_mission wrote:If \(x\) and \(y\) are positive integers and \((5^x)−(5^y)=(2^{y−1})\cdot (5^{x−1}),\) what is the value of \(xy\)?

A. 48
B. 36
C. 24
D. 18
E. 12

[spoiler]OA=E[/spoiler]

Source: Manhattan GMAT

Hi Gmat_mission.

Let's start rewriting the given equation as follows: $$5^x-5^y=2^{y-1}5^{x-1}$$ $$5^x-2^{y-1}5^{x-1}=5^y$$ $$5^x\left(1-2^{y-1}5^{-1}\right)=5^y$$ $$5^x\left(1-\frac{2^y}{2}\cdot\frac{1}{5}\right)=5^y$$ $$5^x\left(\frac{10-2^y}{10}\right)=5^y$$ Now, since the right hand side is always positive, we have that the expression between the parenthesis must be positive, which is possible only when \(y=1,2,3\) (remember that \(x\) and \(y\) are positive integers).

By trial an error, we can see that only \(y=3\) will imply that \(x\) is an integer. Then, let's find \(x\):

If \(y=3\) then $$5^x\left(\frac{10-2^3}{10}\right)=5^3$$ $$5^x\left(\frac{2}{10}\right)=5^3$$ $$5^x\left(\frac{1}{5}\right)=5^3$$ $$5^{x-1}=5^3$$ $$x-1=3$$ $$x=4.$$ Finally, we get that \(xy=4\cdot 3=12\).

Hence, the correct answer is the option _E_.

I hope it helps you. <i class="em em---1"></i>