2009 AMC 12B Problem 5

Below is the professionally curated solution for Problem 5 of the 2009 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12B solutions, or check the answer key.

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Concepts:prime factorization power of 2

Difficulty rating: 1080

Kiana has two older twin brothers. The product of their three ages is $128.$ What is the sum of their three ages?

$10$

$12$

$16$

$18$

$24$

Solution:

Since $128 = 2^7,$ each age is a power of $2.$ The twins share an age $t,$ so Kiana's age is $\dfrac{128}{t^2}.$

Taking $t = 8$ gives Kiana $\dfrac{128}{64} = 2,$ who is younger than the twins. (Smaller twins would make Kiana older, which is not allowed.) The sum is $8 + 8 + 2 = 18.$

Thus, the correct answer is D.

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