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2009 AMC 12B Problem 5

Problem 5 of 25EasierNumber Theory

Kiana has two older twin brothers. The product of their three ages is 128.128. What is the sum of their three ages?

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Solution

Since 128=27,128 = 2^7, each age is a power of 2.2. The twins share an age t,t, so Kiana’s age is 128t2.\dfrac{128}{t^2}. Taking t=8t = 8 gives Kiana 12864=2,\dfrac{128}{64} = 2, who is younger than the twins. (Smaller twins would make Kiana older, which is not allowed.) The sum is 8+8+2=18.8 + 8 + 2 = 18. Thus, the correct answer is D.

More practice

Concepts: prime factorization · power of 2

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