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2009 AMC 10A Problem 13

Problem 13 of 25IntermediateNumber TheoryArithmetic

Suppose that P=2mP = 2^m and Q=3n.Q = 3^n. Which of the following is equal to 12mn12^{mn} for every pair of integers (m,n)?(m, n)?

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Solution

Since 12=22⋅3,12 = 2^2 \cdot 3, 12mn=22mn⋅3mn=(2m)2n⋅(3n)m=P2nQm. \begin{aligned} 12^{mn} &= 2^{2mn} \cdot 3^{mn} \\ &= (2^m)^{2n} \cdot (3^n)^m \\ &= P^{2n} Q^m. \end{aligned} Thus, the correct answer is E.
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Tagged: exponent · prime factorization

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