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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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