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12 Days of Christmas GMAT Competition - Day 2: P is a positive integer 15 Dec 2021, 07:00
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12 Days of Christmas GMAT Competition with Lots of Fun

P is a positive integer. How many positive factors does P have?

(1) P/3 is a prime number.
(2) P/2 is a prime number.



 


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C
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12 Days of Christmas GMAT Competition - Day 2: P is a positive integer 15 Dec 2021, 07:09
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Bunuel
12 Days of Christmas GMAT Competition with Lots of Fun

P is a positive integer. How many positive factors does P have?

(1) P/3 is a prime number.
(2) P/2 is a prime number.





 


This question was provided by GMAT Club
for the 12 Days of Christmas Competition

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Given: P is +ve integer.
To find: +ve factors of P.

(1) P/3 is a prime number.
P can be 6 in that case number will be 2(prime)
P can be 9 in that case number will be 3(prime)
Prime multilpes of 3= 6, 9, 15 , 21 and so on.
No unique answer is possible
Not sufficient. A and D are Out

(2) P/2 is a prime number.
Prime multiples of 2 = 4(with 2), 6(with 3), 10 (with 5) and so on.
No unique answer possible. B is out.

Combining only possible answer is 6. We can find the factors of 6. Sufficient C is the answer.
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12 Days of Christmas GMAT Competition - Day 2: P is a positive integer 16 Dec 2021, 08:00
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Bunuel
12 Days of Christmas GMAT Competition with Lots of Fun

P is a positive integer. How many positive factors does P have?

(1) P/3 is a prime number.
(2) P/2 is a prime number.



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12 Days of Christmas GMAT Competition - Day 2: P is a positive integer 15 Dec 2021, 07:13
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st1: p/3 gives a prime number but then p can be 6 and as well as 9. so if p is 6 then it has 4 factors (1,2,3,6). but if p is 9 then it has 3 factors (1,3,9).
Not sufficient.
st2: p/2 gives a prime number but then p can be 6 and as well as 4. so if p is 6 then it has 4 factors (1,2,3,6). but if p is 4 then it has 3 factors (1,2,4).
Not sufficient.
Combining 1+2: this leaves us with only 6 so the number of factors is 4. Sufficient.
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12 Days of Christmas GMAT Competition - Day 2: P is a positive integer 15 Dec 2021, 07:13
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The number of factors of the positive integer P will depend on the number of prime factors that result in P.

Statement 1

\(\frac{P}{3}\) is prime.

So, we know that P has at least one 3, however we don't know the other number.

For example, if P = 9 i.e. 3*3, the number of positive factors will be 3
However, if P = 3 * (Some other prime number), the number of positive factors will be 4

Hence, the information is not sufficient.

Statement 2

\(\frac{P}{2}\) is prime.

So, we know that P consists of at least one 2, however we don't know the other number.

For example, if P = 4 i.e. 2*2, the number of positive factors will be 3
However, if P = 2 * (Some other prime number), the number of positive factors will be 4.

Hence, the information is not sufficient.

Combining Statement 1 & 2

We know that P has two prime factors, i.e. 2 & 3, hence the total number of positive factors is 4.

Hence sufficient.

IMO - C
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12 Days of Christmas GMAT Competition - Day 2: P is a positive integer 15 Dec 2021, 16:16
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Question Stem:
P is a positive integer. How many positive factors does P have?
(1) P/3 is a prime number.
(2) P/2 is a prime number.

Solution:

P is a positive integer. P>0. We need to find the value of P to ascertain the number of factors of P.

Statement 1: P/3 is a prime number.

Now, a prime number has only 2 factors, 1 and the number itself.
P/3= prime number(say P1)
P=3P1
IF P1 is any prime number except 3,
we have P= multiplication of 2 different prime numbers eg P= 3*2, P=3*5, P=3*7... Thus, we can easily find the number of positive factors of P which is equal to 4.
But if, P1=3, we have P=3*3. Then, the number of factors of P is 3.
Hence, we are getting 2 different values. Statement 1 is not sufficient.

Statement 2: P/2 is a prime number.

Here, we have a similar situation as in Statement 1, we have P=2*another prime(say P2). This will lead to a similar result. Thus, Statement 2 is also not sufficient.

Statement 1+ Statement 2:

From 1, P=3*P1
From 2, P=2*P2
we have, 3*P1= 2*P2
P1=(2/3)*P2
Now, P1 and P2 are integers. P2 should be a prime number divisible by 3. There is only 1 value for P2=3.
Thus, P1=2.
Therefore, P=3*P1=2*P2=6.
Since we have a unique value for P, we can find the number of factors.
The answer is C.
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12 Days of Christmas GMAT Competition - Day 2: P is a positive integer 19 Aug 2024, 10:25
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