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If n/2 is the cube of a positive integer and n/3 is the square of

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Bunuel
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If n/2 is the cube of a positive integer and n/3 is the square of [#permalink] New post   07 Jul 2023, 11:45
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If \(\frac{n}{2}\) is the cube of a positive integer and \(\frac{n}{3}\) is the square of a positive integer. What is the minimum possible value of \(n\)?

A. 423
B. 432
C. 442
D. 447
E. 532
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B
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Re: If n/2 is the cube of a positive integer and n/3 is the square of [#permalink] New post   07 Jul 2023, 14:20
Bunuel wrote:
If \(\frac{n}{2}\) is the cube of a positive integer and \(\frac{n}{3}\) is the square of a positive integer. What is the minimum possible value of \(n\)?

A. 423
B. 432
C. 442
D. 447
E. 532


n should be an even number. Hence, we can eliminate A and D.

Among the other options, let's start with Option B as we are looking for the minimum value of n.

\(432 = 2^4 * 3^3\)

  • \(\frac{432 }{ 2} = 2^3 * 3^3 = 6^3\)

  • \(\frac{432 }{ 3} = 2^4 * 3^2 = 12^2\)

Therefore 432 satisfies both the conditions mentioned in the question.

Option B
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Re: If n/2 is the cube of a positive integer and n/3 is the square of [#permalink] New post   07 Jul 2023, 21:15
gmatophobia how did we determine that n is even ?
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Re: If n/2 is the cube of a positive integer and n/3 is the square of [#permalink] New post   07 Jul 2023, 21:25
AnujL wrote:
gmatophobia how did we determine that n is even ?


AnujL refer to this statement -

\(\frac{n}{2}\) is the cube of a positive integer.


Inference:
1) n must have a 2 in the first place for us to perform the division operation; The result of the operation \(\frac{ n}{2}\) is an integer, so n = some integer * 2
2) If n = some integer * 2, n is even.

Hope this helps !
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Re: If n/2 is the cube of a positive integer and n/3 is the square of [#permalink] New post   08 Jul 2023, 10:27
given n/2 is cube and n/3 is square
so n has to be even and should be divisible by 3
432 = sum of digits is divisible by 3
sufficient to condition as well and minimum value
432 is correct answer option B

Bunuel wrote:
If \(\frac{n}{2}\) is the cube of a positive integer and \(\frac{n}{3}\) is the square of a positive integer. What is the minimum possible value of \(n\)?

A. 423
B. 432
C. 442
D. 447
E. 532
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Re: If n/2 is the cube of a positive integer and n/3 is the square of [#permalink]
08 Jul 2023, 10:27
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