If $n$ is a positive integer and $n^3$ is a multiple of $550,$ what is the least possible value of $n?$

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Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pmFollowed by: 5 members; Thanked: 6 times

by Vincen » Wed Jul 22, 2020 12:21 pm

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If $n$ is a positive integer and $n^3$ is a multiple of $550,$ what is the least possible value of $n?$

A. 8
B. 27
C. 81
D. 110
E. 125

[spoiler]OA=D[/spoiler]

Source: Veritas Prep

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Source: Beat The GMAT — Problem Solving | Wed Jul 22, 2020 12:21 pm

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8176; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: If $n$ is a positive integer and $n^3$ is a multiple of $550,$ what is the least possible value of $n?$

by Scott@TargetTestPrep » Sat Jul 25, 2020 2:22 pm

Vincen wrote: ↑
Wed Jul 22, 2020 12:21 pm
If $n$ is a positive integer and $n^3$ is a multiple of $550,$ what is the least possible value of $n?$

A. 8
B. 27
C. 81
D. 110
E. 125

[spoiler]OA=D[/spoiler]

Solution:

Let’s prime factorize 550:

550 = 55 x 10 = 5 x 11 x 5 x 2 = 2 x 5^2 x 11

Since n^3 must have its prime factors in multiplicities of 3, the least possible value of n^3 is 2^3 x 5^3 x 11^3 (i.e., n = 550 x 2^2 x 5 x 11^2) and hence the least possible value of n is 2 x 5 x 11 = 110.

Answer: D

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