2004 AMC 12B Problem 3

Problem 3 of 25EasierAlgebra Number Theory

x

and

y

are positive integers for which

2^x 3^y = 1296,

what is the value of

x + y?

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Solution

Factoring,

1296 = 6^4 = 2^4 \cdot 3^4.

Matching exponents gives

x = 4

and

y = 4,

x + y = 8.

Thus, the correct answer is A.

Concepts: prime factorization · exponent

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