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

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2012 AMC 10B Problem 10

Below is the professionally curated solution for Problem 10 of the 2012 AMC 10B, 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 of the real AMC 8, AMC 10, AMC 12, and AIME problems in our complete solution collection are used with official legal permission of the Mathematical Association of America (MAA).

Concepts:factor counting prime factorization

Difficulty rating: 960

10.

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

$6$

$7$

$8$

$9$

$10$

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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Problem 10 in Other Years

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