2009 AMC 10B Problem 6

Problem 6 of 25EasierAlgebra Number Theory

Kiana has two older twin brothers. The product of their three ages is

128.

What is the sum of their three ages?

Show solution

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,

j\le3.

Because Kiana is younger than the twins,

7-2j\lt j,

j\ge3.

Therefore

j=3,

giving

t=8

and

k=2.

The sum is

8+8+2=18.

Thus, the correct answer is D.

Tagged: prime factorization · power of 2 · ages