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X is divisible by 144.

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Junior | Next Rank: 30 Posts
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by szy » Sun Jun 20, 2010 4:26 pm
X is divisible by 144. If the cubed root of X is an integer, then which of the following is the cubed root of x definitely divisible by? Choose all that apply.

a) 4
b) 8
c) 9
d) 12


Answer: 4 and 12


Prime factorization of 144: 2^4 * 3^2

The explanation for this problem says that in order for the cube root of a number to be an integer, the original number must have 3 of each prime factor or some multiple of 3. So, that means the factors of X that we can't see must include at least two additional 2's and one additional 3. That's what I don't understand. Why couldn't it be one additional 2 and two additional 3s? Can someone explain this to me pleaaase?
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by hardik.jadeja » Sun Jun 20, 2010 5:35 pm
This isnt a GMAT question. On GMAT you dont have to select multiple answers. But its a good question.

Anyways, here you go..

See if x is divisible by 144 (2^4*3^2) that means 2^4*3^2 a is definitely factor of x. We are also told that cubed root of X is an integer. This means that X is a perfect cube. In order to be a perfect cube, powers (in this case 4 and 2) of the prime factors(in this case 2 and 3) of X must be divisible by 3. 4 and 2 are not divisible by 3. To make them divisible by 3, we increase 4 by 2, that gives are 6. Similarly we increase 2 by 1, that give us 3.

So one of the possible values of X could be 2^6*3^3 = 1728. Observe that 1728 is a cube of 12. So cube root of 1728 is 12 and 12 is divisible by 4 and 12. Thats the answer.

Hope that helps...
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by szy » Sun Jun 20, 2010 6:40 pm
Yeah this question was on one of the MGMAT flashcards for reviewing concepts. Thanks so much for breaking it down and explaining this - makes a lot of sense now!
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by Testluv » Mon Jun 21, 2010 2:22 am
Other way (no prime factorizing and no multiplication):

x is divisble by 144. Thus, the smallest value of x is 12*12.

The cube root of x is an integer. Thus, x is a perfect cube. Thus, the smallest possible value now is 12*12*12.

The cube root of 12*12*12 is 12.

12 is divisble by 12 and by 4 but not by 8 and not by 9.
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