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# If p is positive, is p prime?

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Math Expert
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If p is positive, is p prime?  [#permalink]

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13 Dec 2016, 11:13
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Difficulty:

65% (hard)

Question Stats:

26% (01:47) correct 74% (01:35) wrong based on 196 sessions

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If p is positive, is p prime?

(1) p^3 has exactly 4 distinct factors
(2) p^2 – p – 6 = 0.

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Re: If p is positive, is p prime?  [#permalink]

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13 Dec 2016, 11:23
1
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

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Re: If p is positive, is p prime?  [#permalink]

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17 Oct 2017, 02:28
1
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.

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Re: If p is positive, is p prime?  [#permalink]

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24 Oct 2017, 12:46
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?
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Re: If p is positive, is p prime?  [#permalink]

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01 Nov 2017, 22:12

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.
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Re: If p is positive, is p prime?  [#permalink]

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Updated on: 05 Nov 2017, 13:12
ToraTora wrote:

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.

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

sarahfiqbal wrote:
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.

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.

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?

Look below.

Originally posted by BlueOwl on 05 Nov 2017, 12:56.
Last edited by BlueOwl on 05 Nov 2017, 13:12, edited 2 times in total.
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Re: If p is positive, is p prime?  [#permalink]

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05 Nov 2017, 13:06
Bunuel wrote:
If p is positive, is p prime?

(1) p^3 has exactly 4 distinct factors
(2) p^2 – p – 6 = 0.

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.

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Re: If p is positive, is p prime?  [#permalink]

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21 Nov 2017, 16:03
1
1
Bunuel wrote:
If p is positive, is p prime?

(1) p^3 has exactly 4 distinct factors
(2) p^2 – p – 6 = 0.

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

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Re: If p is positive, is p prime?  [#permalink]

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24 Nov 2017, 13:50

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

Re: If p is positive, is p prime? &nbs [#permalink] 24 Nov 2017, 13:50
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