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# If p, q, x, and y are positive integers, is q^y/p^x an integer?

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Math Expert
Joined: 02 Sep 2009
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If p, q, x, and y are positive integers, is q^y/p^x an integer?  [#permalink]

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12 Nov 2014, 09:46
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77% (01:04) correct 23% (01:13) wrong based on 140 sessions

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Tough and Tricky questions: Number Properties.

If $$p$$, $$q$$, $$x$$, and $$y$$ are positive integers, is $$\frac{q^y}{p^x}$$ an integer?

(1) $$q$$ is evenly divisible by $$p$$

(2) $$y \ge x$$

Kudos for a correct solution.

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Joined: 29 Sep 2014
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Re: If p, q, x, and y are positive integers, is q^y/p^x an integer?  [#permalink]

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12 Nov 2014, 10:14
2
(1) if p divides q then q = k x p with k positive integer
q^y / p^x = k^y x p^(y-x)
so the question boils down to the following: is p^(y-x) an integer?
We are clearly missing information here, so INSUFFICIENT. (3^(-1) is not an integer but 3^1 is).

(2) Clearly INSUFFICIENT

(1) + (2) tells us that p^(y-x) is an integer. SUFFICIENT.

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Re: If p, q, x, and y are positive integers, is q^y/p^x an integer?  [#permalink]

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12 Nov 2014, 21:05
1
Statement 1: q is evenly divisible by p

Let say q=p=2 and x = 4, y =2 (i.e y < x )
=> q^y/p^x = 2^2/2^4 = No an integer .... (1)

But q=p=2 and x = 4, y =4 (ie y >= x)
=> q^y/p^x = 2^4/2^4 = It is an integer .... (2)

Therefore, insufficient

Statement 2 : y > = x,
Insufficient, since we don't if q is a multiple of p or not

Combining 1 and 2: Sufficient, since it corresponds to only the (2) case.

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Re: If p, q, x, and y are positive integers, is q^y/p^x an integer?  [#permalink]

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12 Nov 2014, 23:27
2
Bunuel wrote:

Tough and Tricky questions: Number Properties.

If $$p$$, $$q$$, $$x$$, and $$y$$ are positive integers, is $$\frac{q^y}{p^x}$$ an integer?

(1) $$q$$ is evenly divisible by $$p$$

(2) $$y \ge x$$

Kudos for a correct solution.

St 1 says q/p= Integer, I so we have q=p*I Subsituting in given eqn we get that (P*I)^y/p^x...clearly we need to know y and x..consider x=100 and p=2,q=8 and y=2...We will get an integer but if x=1,y=2 then an integer...not sufficient.. A and D ruled out

2. says $$y \ge x$$ but we don't know whether p is factor of q or not.. B is out

Combining, both are conditions are met so Ans is C and we will get an Integer
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Joined: 02 Sep 2009
Posts: 58398
Re: If p, q, x, and y are positive integers, is q^y/p^x an integer?  [#permalink]

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13 Nov 2014, 09:02
1
Bunuel wrote:

Tough and Tricky questions: Number Properties.

If $$p$$, $$q$$, $$x$$, and $$y$$ are positive integers, is $$\frac{q^y}{p^x}$$ an integer?

(1) $$q$$ is evenly divisible by $$p$$

(2) $$y \ge x$$

Kudos for a correct solution.

Official Solution:

The question asks if we can determine whether the expression $$\frac{q^y}{p^x}$$ yields an integer. This is the same as asking if $$q^y$$ is divisible by $$p^x$$ or, equivalently, if $$p^x$$ is a factor of $$q^y$$.

Statement 1 tells us that $$q$$ is evenly divisible by $$p$$, or, in other words, that $$p$$ is a factor of $$q$$. Let us plug in numbers that satisfy this condition and see if we can answer the question in the prompt. We will choose $$p = 2$$ and $$q = 4$$. However, since we are given no information on $$x$$ or $$y$$, we can plug in any integer values for $$x$$ and $$y$$. If $$x = 3$$ and $$y = 1$$, then $$\frac{q^y}{p^x} = \frac{4^1}{2^3} = \frac{4}{8} = \frac{1}{2}$$, which is not an integer. On the other hand, if $$y = 3$$ and $$x = 1$$, $$\frac{q^y}{p^x} = \frac{4^3}{2^1} = \frac{64}{2} = 32$$, which is an integer. Statement 1 alone is NOT sufficient to answer the question. Eliminate answer choices A and D. The correct answer choice is B, C, or E.

Statement 2 tells us that $$y \ge x$$. However, we know nothing about the values of $$p$$ and $$q$$, which can be any integers. Whether the expression yields an integer depends on the relationship between $$p$$ and $$q$$. Statement 2 alone is also NOT sufficient to answer the question. Eliminate answer choice B. The correct answer choice is either C or E.

Taking both statements together: Statement 1 tells us that $$p$$ is a factor of $$q$$, which implies that $$p \le q$$, and statement 2 tells us that $$x \le y$$. This means that the term $$q^y$$ will always be larger than $$p^x$$, and since $$p$$ is a factor of $$q$$, $$\frac{q^y}{p^x}$$ will always yield an integer. For example, if $$p = 3$$, $$q = 6$$, $$x = 1$$, and $$y = 2$$, then $$\frac{q^y}{p^x} = \frac{6^2}{3^1} = \frac{36}{3} = 12$$. Both statements together are sufficient.

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Re: If p, q, x, and y are positive integers, is q^y/p^x an integer?  [#permalink]

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25 Jan 2018, 05:23
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Re: If p, q, x, and y are positive integers, is q^y/p^x an integer?   [#permalink] 25 Jan 2018, 05:23
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