Question 7
There are positive integers $x$ and $y$ that satisfy the system of equations $$
\log_{10} x + 2\log_{10}(\gcd(x, y)) = 60
$$ $$
\begin{aligned} &\log_{10} y \\ &\quad {}+ 2\log_{10}(\operatorname{lcm}(x, y)) = 570. \end{aligned}
$$ Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x,$ and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y.$ Find $3m + 2n.$
\log_{10} x + 2\log_{10}(\gcd(x, y)) = 60
$$ $$
\begin{aligned} &\log_{10} y \\ &\quad {}+ 2\log_{10}(\operatorname{lcm}(x, y)) = 570. \end{aligned}
$$ Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x,$ and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y.$ Find $3m + 2n.$