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Math Expert V
Joined: 02 Sep 2009
Posts: 56300
If x, y, p and q are positive integers, is x^p a factor of y^q?  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 76% (01:37) correct 24% (01:42) wrong based on 61 sessions

### HideShow timer Statistics If x, y, p and q are positive integers, is x^p a factor of y^q?

(1) x is a factor of y.
(2) p < q + 1

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Math Expert V
Joined: 02 Aug 2009
Posts: 7764
Re: If x, y, p and q are positive integers, is x^p a factor of y^q?  [#permalink]

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If x, y, p and q are positive integers, is x^p a factor of y^q?

(1) x is a factor of y.
There are 4 variables but we know relationship of just two, so not possible to have a definite answer.
If x is 2, y is 4..
A) p is 100 and q is 2..... x^p is NOT a factor of y^q
B) but if p is 2 and q is 100, YES
Insufficient

(2) p < q + 1
Insufficient

Combined..
x is a factor of y and p is not greater than a..
x^p will be atleast equal to y^q or a factor of it..
Sufficient

C
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Re: If x, y, p and q are positive integers, is x^p a factor of y^q?  [#permalink]

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Bunuel wrote:
If x, y, p and q are positive integers, is x^p a factor of y^q?

(1) x is a factor of y.
(2) p < q + 1

For x^p to be a factor of y^q, (y^q)/(x^p) must be an integer. This means if we were to break y and x into their prime factors, y must be a multiple of x and q must be greater than or equal to p.

This should make logical sense if you understand your basic divisibility rules and how to cancel out numbers when dividing. Some examples to illustrate:

Example 1: (3^5)/(3^2) = 3^3 = 27(an integer).
Example 2: (9^2)/(3^5) = (3^4)/(3^5) = 1/3(not an integer).

In the second case, note that how even though x is a multiple of y, q is less than p and therefore we don't get an integer.

Statement 1) Using our examples from above we see that in each case x is a factor of y. However in example 1 we get an integer and in example we do not get an integer.

INSUFFICIENT

Statement 2) Since p & q must be integers, this tells us q is greater than or equal to p. This is part of what we're looking for, however we still need to know if y is a multiple of x.

INSUFFICIENT

Statements 1 & 2) Now we know that y is a multiple of x and q is greater than or equal to p. Exactly what we're looking for.

SUFFICIENT

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Unlimited private GMAT Tutoring in Chicago for less than the cost a generic prep course. No tracking hours. No watching the clock. Re: If x, y, p and q are positive integers, is x^p a factor of y^q?   [#permalink] 24 Sep 2018, 11:43
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