Divisibility

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Divisibility

by Franny » Mon Apr 29, 2013 10:12 am
Hey guys,

I don't really understand the bold part of the answer below. Can anyone please explain!

Question:
x is divisible by 144. If ³√x is an integer, then which of the following
is ³√x definitely divisible by? (Choose all that apply)

Answer:
Remember that when we complete a prime box for a variable, that
variable could still have additional factors. 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 (3, 6, 9, etc.). In this case, that
means the factors of x that we can't see must include at least two
additional 2s and one additional 3
. From this information, we can
definitively conclude that ³√x must have two 2s and one 3 as
factors. 4 and 12 are the only numbers in the list we can guarantee
are factors of ³√x .
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by Anju@Gurome » Mon Apr 29, 2013 10:42 am
Franny wrote:x is divisible by 144. If ³√x is an integer, then which of the following is ³√x definitely divisible by?
As ³√x is an integer, x must be a cube of an integer.
Let us assume, x = 144k = (12²)*k, where k is some positive integer.
We can see that minimum possible value of k such that (12²)*k becomes a cube of an integer is 12.
Hence, minimum possible value of x is 12³
So, minimum possible value of ³√x is 12.

Hence, ³√x will be definitely divisible by any factor of 12.

A discussion on some similar problems
https://www.beatthegmat.com/how-to-solve ... 16927.html
https://www.beatthegmat.com/if-p-is-a-po ... tml#628225
Anju Agarwal
Quant Expert, Gurome

Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.

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