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# If px = qy, is x/y an integer?

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If px = qy, is x/y an integer? [#permalink]

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08 May 2013, 08:13
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If px = qy, is x/y an integer?

(1) q and p are primes.
(2) Positive integer y is neither composite nor prime

Question created by me.
[Reveal] Spoiler: OA

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Last edited by mau5 on 19 May 2013, 07:49, edited 1 time in total.

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Re: If px = qy, is x/y an integer? [#permalink]

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08 May 2013, 08:28
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If px = qy, is x/y an integer?

I.q and p are primes.
$$3*20=2*30$$ => $$\frac{20}{30}$$ no integer
$$3*1=3*1$$ => $$\frac{1}{1}$$ integer
Not sufficient

II. Positive integer y is neither composite nor prime
So y is $$1$$, no info about x.
Not sufficient

I + II)$$px=q$$, if $$p=q$$ then $$x=1$$ and $$\frac{x}{y}=\frac{1}{1}$$ is an integer
if $$p\neq{q}$$ ie: $$5*x=7, x=\frac{7}{5}$$, then $$\frac{x}{y}$$ is NOT an integer
Not sufficient.
E
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Re: If px = qy, is x/y an integer? [#permalink]

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08 May 2013, 09:22
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Expert's post
vinaymimani wrote:
If px = qy, is x/y an integer?

I.q and p are primes.
II. Positive integer y is neither composite nor prime

Question: is x/y an integer?

px = qy
x/y = q/p

We need to know whether q/p is an integer.

I.q and p are primes.
This means that q has only two factors - 1 and q itself. If p = q, then p is a factor q, else it is not.
Not sufficient.

II. Positive integer y is neither composite nor prime
This means y = 1. We still cannot say whether x/y is an integer since x may or may not be an integer.
Not sufficient.

Both taken together, nothing changes. We still do not know whether x/y is an integer.
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Re: If px = qy, is x/y an integer? [#permalink]

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20 Dec 2015, 16:36
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Re: If px = qy, is x/y an integer? [#permalink]

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21 Dec 2015, 22:47
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If px = qy, is x/y an integer?

(1) q and p are primes.
(2) Positive integer y is neither composite nor prime

In the original condition, there are 4 variables(pq,x,y) and 1 equations(px=qy), which should match with the number of equations. So you need 3 more equations. For 1) 1 equation, for 2) 1 equation, which is likely to make E the answer. In 1) & 2), from p=q=2, x/y=1, which is yes. From p=5, q=2, x=1, y=2.5, which is no. So it is not sufficient. Therefore, the answer is E.

-> For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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Re: If px = qy, is x/y an integer?   [#permalink] 21 Dec 2015, 22:47
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