How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}{3}$ are $4$-digit integers?

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M7MBA: Moderator; Posts: 2058; Joined: Sun Oct 29, 2017 4:24 amFollowed by: 5 members; Thanked: 1 times

by M7MBA » Wed May 20, 2020 7:37 am

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How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}{3}$ are $4$-digit integers?

A. $111$
B. $112$
C. $333 $
D. $334$
E. $1,134$

[spoiler]OA=B[/spoiler]

Source: Official Guide

Source: Beat The GMAT — Problem Solving | Wed May 20, 2020 7:37 am

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8246; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}{3}$ are $4$-digit integers?

by Scott@TargetTestPrep » Mon May 25, 2020 10:51 am

M7MBA wrote: ↑
Wed May 20, 2020 7:37 am
How many positive integers $n$ have the property that both $3n$ and $\dfrac{n}{3}$ are $4$-digit integers?

A. $111$
B. $112$
C. $333 $
D. $334$
E. $1,134$

[spoiler]OA=B[/spoiler]

Source: Official Guide

Solution:

Since n/3 is an integer, n must be a multiple of 3. The largest multiple of 3 that contains the mentioned properties is 3333, and the smallest is 3000. The number of multiples of 3 from 3000 to 3333 is:

(3333 - 3000)/3 + 1 = 333/3 + 1 = 112.

Answer: B

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