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

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)

Answer: (E)

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.

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

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.

Answer: E

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!
Aravind BT

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