Finding the Integers #2

Let $a=gx$ and $b=gy$ where $g=gcd(a,b)$ . It can be easily seen that $gy$ $\leqslant$ $5gx$ . The equation becomes $g^{5x-y}x^{5x}=y^y$ . Thus $x^{5x}|y^y$ ; implying $x=1$ . So we get $g^{5-y}=y^y$ . Now $y$ $\leqslant$ 5. On inspection only $y=4$ satisfies the required condition. At last $a=4^4$ and $b=4^5$ .