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# If p and q are positive integers and pq = 24, what is the

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Re: If p and q are positive integers and pq = 24, what is the [#permalink]
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If p and q are positive integers and pq = 24, what is the value of p ?

(1) q/6 is an integer.
(2) p/2 is an integer.

Given
$$pq = 24$$
$$p$$ and $$q$$are positive integers: $$p > 0$$ , $$q > 0$$

Statement 1

$$q/6$$ is an integer => $$q = 6k$$, k is a positve integer as $$q > 0$$

Substituting $$q$$ in $$pq = 24$$, we obtain $$p*6k =24$$.
As $$6k > 0$$, dividing both sides by$$6k$$, we obtain $$p = 24/(6k) = 4/k$$.
$$p$$ can be $$4$$, $$2$$ and $$1$$ for $$k= 1$$, $$2$$ and $$4$$.

As$$p$$ can not be determined uniquely, statement (1) is not sufficient....................................(A), (D)

Statement 2

$$p/2$$ is an integer, $$p = 2l$$, $$l$$is a positve integer as $$p > 0$$
Substituting $$p$$ in the equation $$pq = 24$$, we obtain $$2l.q =24$$.
As $$2l > 0$$, dividing both sides by$$2l$$, we obtain $$q = 24/(2l) = 12/l$$.
$$q$$ can be $$12$$, $$6$$,$$4$$,$$3$$,$$2$$ and $$1$$ for $$k= 1$$, $$2$$, $$3$$, $$4$$,$$6$$ and $$12$$.
Or,$$p$$ can be $$2$$,$$4$$,$$6$$,$$8$$ and $$12$$ in order to satisfy$$p*q = 24$$

As$$p$$ can not be determined uniquely, statement (2) is not sufficient....................................(B)

Combining statements (1) and (2), $$p*6k = 2l*q$$
Substituting $$3p = q$$ in equation$$pq=24$$,
Or, $$3p*p=24$$
Or,$$p =\sqrt{8}$$
=> $$p$$ is not an integer;thus, both statements (1) and (2) are not sufficient........................(C)

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Re: If p and q are positive integers and pq = 24, what is the [#permalink]
Bunuel
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If p and q are positive integers and pq = 24, what is the value of p ?

(1) q/6 is an integer.
(2) p/2 is an integer.

Data Sufficiency
Question: 45
Category: Arithmetic Arithmetic operations
Page: 156
Difficulty: 600

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I solved this question using prime factorization box and got correct answer. But i think that approach is not reasonable for these kind of questions because there could only one single pair which could satisfy both statements. ( even in that case, prime factorization box would still recommend that there is no definitive answer because we dont know all numbers present in prime box). Kindly correct me if i am wrong.
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Re: If p and q are positive integers and pq = 24, what is the [#permalink]
pq = 24
possible values for p and q
1*24 = 24
2*12 = 24
3*8 = 24
4*6 = 24
6*4 = 24
8*3 = 24
12*2 = 24
24*1 = 24

from stat 1 -
possible values of p when q/6 is an integer
1*24 = 24
2* 12 = 24
4*6 = 24

p can be 1 or 2 or 4 .not sufficient.
from stat 2
when p/2 is and integer then possible values of p itself -
2*12
4*6
6*4
12*2
24*1
p can be 2 , 4, 6, 12, 24. not sufficient .

now from 1 + 2
possible values of p when q/6 and p/2 are integers
2*12
4*6
p can be either 2 or 4. not sufficient
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Re: If p and q are positive integers and pq = 24, what is the [#permalink]
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Bunuel
If p and q are positive integers and pq = 24, what is the value of p ?

(1) q/6 is an integer.
(2) p/2 is an integer.

Solution:

Question Stem Analysis:

We need to determine the value of p, given that pq = 24 and both p and q are positive integers. We see that p (and q) are factors of 24. Therefore, p (and q) can be one of the following integers:

1, 2, 3, 4, 6, 8, 12, and 24

Statement One Alone:

Since q/6 is an integer, q is 6, 12, or 24, and p will then be 4, 2, or 1, respectively. Since p can take on more than one value, statement one alone is not sufficient.

Statement Two Alone:

Since p/2 is an integer, p is 2, 4, 6, 8, 12, or 24. Since p can take on more than one value, statement two alone is not sufficient.

Statements One and Two Together:

From the two statements, we see that p can still be either 2 or 4. Since p can take on more than one value, both statements are not sufficient.

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Re: If p and q are positive integers and pq = 24, what is the [#permalink]
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Since pq = 24 and p and q are positive integers, we can conclude that p and q are positive integral factors of 24. The positive integral factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

From statement I alone, q/6 is an integer.

Therefore, q is a multiple of 6. Remember that q is also a factor of 24. Therefore, the possible values for q = 6 or 12 or 24. And because the product of p ad q is 24, there will be 3 corresponding values for p.
Statement I alone is insufficient. Answer options A and D can be eliminated.

From statement II alone, p/2 is an integer.

This means that p is an even factor of 24 and hence p can be any of 2, 4, 6, 8, 12 or 24.
Statement II alone is insufficient. Answer option B can be eliminated.

Combining statements I and II, we have the following:

From statement II, p is even; from statement I alone, q = 6 or 12 or 24.
For q = 6, p = 4 and for q = 12, p = 2.
The combination of statements is insufficient to find a unique value for p. Answer option C can be eliminated.

The correct answer option is E.

Hope that helps!
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Re: If p and q are positive integers and pq = 24, what is the [#permalink]
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