2024 AMC 10A Problems/Problem 6

Problem

How many ordered pairs $(m, n)$ of positive integers exist such that $m$ is a factor of $54$ and $mn$ is a factor of $70$?

$\textbf{(A)}~2 \qquad\textbf{(B)}~4 \qquad\textbf{(C)}~10 \qquad\textbf{(D)}~12 \qquad\textbf{(E)}~16$

Solution

The greatest common divisor of $54$ and $70$ is $2$, so we know that $m$ is either $1$ or $2$. If $m = 1$, then $n$ can be any factor of $70$, so $8$ options. If $m=2$, $n$ can be any factor of $35$, so $4$ options. In total, there are $\boxed{\textbf{(D)}~12}$ ordered pairs.

~joshualiu315

See also

2024 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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All AMC 10 Problems and Solutions

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