2012 AMC 10B Problem 10 — Solution | LIVE by Po-Shen Loh

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2012 AMC 10B Problem 10

Attempt Problem 10 of the 2012 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2012 AMC 10B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

10.

How many ordered pairs of positive integers $(M,N)$ satisfy the equation $\frac{M}{6}=\frac{6}{N}?$

$6$

$7$

$8$

$9$

$10$

Answer: D

Concepts:factor counting prime factorization

Difficulty rating: 960

Solution:

By cross multiplying, we can see that $MN = 36.$ Thus, we can make $M$ any factor of $36$ and then determine $N$ from it.

Since $36=2^2\cdot 3^2,$ we have $(2+1)(2+1)=9$ possible choices for $M,$ each of which also determine a unique $N.$

Thus, the correct answer is D .

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