For how many positive integer values of n are both (n)/(3) and 3n three-digit whole numbers?

2008 AMC 8 Problem 22 — Solution

Below is the professionally curated solution for Problem 22 of the 2008 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2008 AMC 8 solutions, or check the answer key.

Difficulty rating: 1340

22.

For how many positive integer values of $n$ are both $\dfrac{n}{3}$ and $3n$ three-digit whole numbers?

$\ 12$

$\ 21$

$\ 27$

$\ 33$

$\ 34$

Let $x=\dfrac{n}{3}$ . Then $n=3x$ , so $3n=9x$ .

Both $x$ and $9x$ must be three-digit whole numbers. Thus $100\le x$ and $9x\le999$ , so $100\le x\le111$ .

There are $111-100+1=12$ integer values of $x$ , and each gives one positive integer value of $n$ .

Thus, A is the correct answer.