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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 \(n?\)

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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 \(n?\)

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]

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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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