If p is positive, is p prime?

Question

If p is positive, is p prime?
 
(1) p^3 has exactly 4 distinct factors
(2) p^2 – p – 6 = 0.

EMPOWERgmatRichC · Accepted Answer

Hi All,

We're told that P is POSITIVE. We're asked if P is PRIME. This is a YES/NO question. To start, it's worth noting that we do NOT know whether P is an integer or not (and if you assume that P must be an integer, then you'll get this question wrong).

1) P^3 has exactly 4 distinct factors.

To start, let's focus on numbers that have just 4 distinct factors. There are two that you should be able to find relatively easily: 
6 (factors are 1, 2, 3 and 6) 
8 (factors are 1, 2, 4 and 8)

According to Fact 1, P^3 could be either of those 2 numbers.... 
IF.... 
P^3 = 6, then P is a NON-INTEGER and the answer to the question is NO 
P^3 = 8, then P = 2 and the answer to the question is YES. 
Fact 1 is INSUFFICIENT

2) P^2 - P - 6 = 0

We can factor this equation into it's pieces and solve... 
P^2 - P - 6 = 0 
(P-3)(P+2) = 0 
P = +3 or -2 
The prompt tells us that P is POSITIVE, so there's only one solution here: +3... and the answer to the question is ALWAYS YES. 
Fact 2 is SUFFICIENT

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

stonecold · Answer

Here's is my solution to this one ->

Given data ->
p>0
We are not told if p is an integer or not.

The question is asking us whether p is prime or not

Statement 1=>
p^3 has 4 factors
p=3 => yes,p is prime
p=(15)^1/3=> p^3=15=> 4 factors => No,p is not prime

Hence not sufficient

Statement 2=>
Here solving the equation =>
 p^2 – p – 6 = 0.
p= 1+5/2 => 3,-2
as p>0 => p must be 3
So,p is prime
Hence sufficient

Hence B

sarahfiqbal · Answer

OK I got this question wrong (I answered B but the OA is D) but I am not far of from grasping this whole integer properties bit. Can someone please clarify:

Statement 2: p^3 has exactly 4 distinct factors

What exactly does this statement mean? What should I be thinking of here? The "4 distinct factors" is not clear to me.

Please please and thank you.

Madhavi1990 · Answer

I get that B is suff. But A? even if P weren't prime say (12)^3 = P^3 = 12 is not prime. On the other hand, (2)^3 says P is prime. So a prime cubed and non prime cubed both will have 4 distinct factors (say 2 cubed has 4,8,2,1). Can someone explain why is A sufficient?

ToraTora · Answer

Answer is D.

For statement 1 p^3 has exactly 4 distinct factors: 1,p,p^2,p^3. Hence prime, therefore sufficient.

For statement 2 when you factorize the equation you get x=2,3. Both are prime, therefore sufficient.

BlueOwl · Answer

For statement 2, you get x = -2, 3.

All "distinct factors" means all the positive, integer factors of a number counted only once. 
Distinct factors of 125 are 1, 5, 25, and 125.

Look below.

BlueOwl · Answer

Let's start with 2, as it's easier.
2) 
Factoring:  p^2 – p – 6 = 0 
Results in:  (p+2)(p-3) = 0 
Therefore:  p = -2  or  +3

Since P is positive, P = 3, 3 is prime.  Sufficient

1) 
The only time  p ^ 3  will have 4 distinct factors is if p is prime.

We can test this: 
If  p = 2, 2 ^ 3 = 8.  The distinct (unique) factors of 8 are: 1, 2, 4, 8
If  p = 3, 3 ^ 3 = 27.  The distinct (unique) factors of 27 are: 1, 3, 9, 27
If  p = 4, 4 ^ 3 = 64.  The distinct (unique) factors of 64 are: 1, 2, 4, 8, 16, 32, 64
If  p = 5, 5 ^ 3 = 125.  The distinct (unique) factors of 125 are: 1, 5, 25, 125
If  p = 6, 6 ^ 3 = 216.  The distinct (unique) factors of 216 are: 1, 2, 3, 4, 6, 8, 9... and so on.

1 is also sufficient.

The answer is D.

hellosanthosh2k2 · Answer

Answer should be B.

The catch is p is positive, but need not be an integer

Statement 1 :  p^3  has 4 distinct factors
 if  p  is prime, then  p^3  has 4 distinct factors
 BUT if  p = x^{1/3} * y ^ {1/3} , where x and y are prime numbers, then  p^3 = (x^{1/3} * y ^ {1/3})^3 = xy  still has 4 distinct factors, in this case p is not prime.

Not Suff

Statement 2: from quad eqn we can p = -2 or 3, since p is positive, p = 3, prime -> Suff

Answer (B)

GMATinsight · Answer

Question : Is p Prime?

Statement 1 : p^3 has exactly 4 distinct factors

Number of factors = (power of 1st prime number+1)*(power of 2nd prime number+1)*(power of 3rd prime number+1)...

i.e.  p^3  will have 4 Factors when p is a Prime number(e.g.  p^3=2^3 ) because (3+1) = 4

BUT if  p = 2^{1/3}*3^{1/3}  then  p^3 = 2*3  and it will still have 4 factors hence

NOT SUFFICIENT

Statement 2 : p^2 – p – 6 = 0

i.e. p = 3 or -2 but given that p is positive hence
p = 3 hence
SUFFICIENT

Answer: Option B

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If p is positive, is p prime?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If p is positive, is p prime?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards