2009 AMC 10B Problem 6

P1-10~2 minPrint

Divisibility & Factors Number Properties Test small/specific values or special cases to find or verify the answer

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

Show solution without answering

Key insight

The twins share an age, so 128 is Kiana's age times the twins' age twice. Try the few possible twin ages and keep the one where Kiana is younger.

Solution

The twins have the same age, so the product is (Kiana's age) $\times$ (twins' age) $\times$ (twins' age). Try each possible twins' age.

$128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ , so every age must be a power of $2$ : $1, 2, 4, 8, \ldots$
List the ways to pick the twins' age $t$ and find Kiana's age from $128 \div (t \times t)$ :

Twins' age	Twins multiplied	Kiana's age	Kiana younger?
$1$	$1$	$128$	no
$2$	$4$	$32$	no
$4$	$16$	$8$	no
$8$	$64$	$2$	yes

(A twins' age of $16$ is too big, since $16 \times 16 = 256$ is more than $128$ .)
3. Only the last row fits the word "older": the twins are $8$ and Kiana is $2$ .
4. The sum is $8 + 8 + 2 = 18$ .

The answer is $\boxed{\textbf{(D)}\ 18}$ .

Why this works

Twins give a repeated factor, so the product has a squared part. Writing $128$ as a product of $2$ 's shows there are only a few possible ages to test. Several choices multiply to $128$ , so the word "older" is what picks the right one; always use every condition in the problem.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Let the twins each be $t$ years old and Kiana $k$ , with $k < t$ . Then $k t^2 = 128 = 2^7$ , so $t$ is a power of $2$ and $t^2$ divides $2^7$ .

$t = 8$ : $k = 128/64 = 2$ , and $2 < 8$ . Valid.
$t = 4$ : $k = 128/16 = 8$ , but then Kiana is older. Invalid.
$t = 2$ or $t = 1$ : $k = 32$ or $128$ , far older. Invalid.

So the ages are $2, 8, 8$ and the sum is $2 + 8 + 8 = 18$ .

The answer is $\boxed{\textbf{(D)}\ 18}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics (§1.4, §1.5)
Introduction to Number Theory, Chapter 5: Divisor Problems (§5.2, §5.3)
Introduction to Algebra, Chapter 20: Special Functions (§20.3)
Introduction to Number Theory, Chapter 13: Divisibility Rules (§13.2)