2019 AIME I Problem 7

There are positive integers x and y that satisfy the system of equations \log_{10} x + 2 \log_{10} (\gcd(x,y)) = 60 \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) = 570. 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.