2012 AMC 10B Problem 10

Problem 10 of 25EasierNumber Theory

How many ordered pairs of positive integers

(M,N)

satisfy the equation

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

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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 .

Concepts: factor counting · prime factorization

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.