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

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

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(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.

Answer (C)
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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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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.

The answer should be C)
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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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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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Re: If p, q, x, and y are positive integers, is q^y/p^x an integer? [#permalink]

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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.


Answer: C.
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New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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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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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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