2008 AMC 8 Problem 22 Number theory

                                    Understand the Problem

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

2008 AMC 8 Problem 22

Topic

Number Theory

Difficulty Level
Easy
Suggested Book
Mathematical Circles

Start with hints

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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