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

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

by Max@Math Revolution » Sun Sep 08, 2019 10:39 pm

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[GMAT math practice question]

n/2 is the cube of a positive integer and 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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by GMATGuruNY » Mon Sep 09, 2019 2:59 am
Max@Math Revolution wrote:[GMAT math practice question]

n/2 is the cube of a positive integer and 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/3 must be a perfect square.
Least possible value for n/3:
n/3 = 2*2
n = 2*2*3 = 12

Implication:
The correct answer must be a multiple of 12 -- and thus must be divisible by both 4 and 3.

For an integer to be multiple of 4, its last two digits must form an integer that can be divided twice by 2.
A. 423 --> 23 CANNOT be divided twice by 2
B. 432 --> 32 can be divided twice by 2
C. 442 --> 42 CANNOT be divided twice by 2
D. 447 --> 47 CANNOT be divided twice by 2
E. 532 --> 32 can be divided twice by 2
Eliminate A, C and D.

For an integer to be a multiple of 3, its digits must sum to a multiple of 3.
B. 432 --> 4+3+2 = 9 --> multiple of 3
E: 532 --> 5+3+2 = 10 --> NOT a multiple of 3
Eliminate E.

The correct answer is B.
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by Max@Math Revolution » Tue Sep 10, 2019 11:42 pm
=>

We have n/2 = a^3 and n/3 = b^2 for some integers a and b.
So, n=2a^3 and n=3b^2 for some integers a and b.
The possible values of n=2a^3 are 2*1^3, 2*2^3, 2*3^3, 2*4^3, 2*5^3, 2*6^3, ... which are 2, 16, 54, 128, 250, 432, ... .
The possible values of n=3a2 are 3*1^2, 3*2^2, 3*3^2, 3*4^2, 3*5^2, 3*6^2, 3*7^2, 3*8^2, 3*9^2, 3*10^2, 3*11^2, 3*12^2,..., which are 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 363, 432, ... .
The first number to appear in both lists is 432.
Thus, the minimum possible value of n is 432.

Therefore, B is the answer.
Answer: B
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