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# Understand the Problem

For how many positive integer values of $$n$$ are both $$\frac {n}{3}$$ and $$3n$$ three-digit whole numbers? $\textbf{(A)}\ 12\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 27\qquad \textbf{(D)}\ 33\qquad \textbf{(E)}\ 34$

#### Number Theory

Easy
##### Suggested Book
Mathematical Circles

Do you really need a hint? Try it first!

Think about the minimum and maximum values of $$\frac{n}{3} \ and \ 3n$$ . Then proceed .
Clearly $$100 \leq \frac{n}{3} \leq 999 \ and \ 100 \leq 3n \leq 999$$ .
As $$\frac {n}{3} \geq 100 \Rightarrow n \geq 300 \ and \ also \ 3n \leq 999 \Rightarrow n \leq 333$$ . So clearly $$300 \leq n \leq 333$$ .
Note that $$\frac{n}{3} \ is \ an \ integer \Rightarrow 3|n$$ . $$\\$$ And we have 12 such values of $$n$$ between 300 and 333 , including 300 and 333 .

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