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

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2024 AMC 10A Problems/Problem 6

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