Question

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

$10$

$12$

$16$

$18$

$24$

Answer: D

Concepts:prime factorization power of 2 ages

Difficulty rating: 960

Solution:

Since $128=2^7,$ every age is a power of $2.$ Writing the twins' common age as $t$ and Kiana's as $k,$ we need $t^2k=128$ with $k\lt t.$

Write $t=2^j.$ Then $k=2^{7-2j}.$ For $k$ to be a positive integer age we need $7-2j\ge0,$ so $j\le3.$ Because Kiana is younger than the twins, $7-2j\lt j,$ so $j\ge3.$ Therefore $j=3,$ giving $t=8$ and $k=2.$ The sum is $8+8+2=18.$

Thus, the correct answer is D.

2009 AMC 10B Problem 6

Answer: D

Difficulty rating: 960

Solution:

Problem 6 in Other Years