2019 AIME I Problems/Problem 7

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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 $3m+2n$.

Solution

One immediately sees that $x = 10^{20}$ and $y = 10^{190}$, so the answer is $3 * 40 + 2 * 380 = \boxed{880}$ because $10^{20} = 2^{20} * 5^{20}$ and similarly for $10^{190}$.

See Also

2019 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
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All AIME Problems and Solutions

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