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.

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10.

How many ordered pairs of positive integers

(M,N)

satisfy the equation

\frac{M}{6}=\frac{6}{N}

6

7

8

9

10

Cross-multiply the equation

Count the positive divisors of

36

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 .

Problem 10 in other years