If p is a positive integer, is p a prime number? (1) p and p

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hazelnut01: Master | Next Rank: 500 Posts; Posts: 157; Joined: Sat Nov 19, 2016 5:34 amFollowed by: 4 members; Thanked: 2 times

by hazelnut01 » Fri May 19, 2017 6:16 am

If p is a positive integer, is p a prime number?

(1) p and p+1 have the same number of factors.
(2) pâˆ’1 is a factor of p.

OA=B

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Source: Beat The GMAT — Data Sufficiency | Fri May 19, 2017 6:16 am

susheelh: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Sat May 06, 2017 10:45 pm

by susheelh » Fri May 19, 2017 9:23 am

In my opinion Answer is B

S1: p and p+1 have the same number of factors.
I could find atleast two cases.
Case 1: P=2,P+1 = 3 -> Both are prime and have two factors
Case 2: P = 14, P+1 = 15 -> Both have 4 factors.
Hence S1 is Insufficient

S2: pâˆ’1 is a factor
only possible case P = 2 and P-1 = 1. Here 2 is prime. Hence one is its factor. Therefore the answer is 2 and Sufficient

ziyuenlau wrote:If p is a positive integer, is p a prime number?

(1) p and p+1 have the same number of factors.
(2) pâˆ’1 is a factor of p.

OA=B

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Fri May 19, 2017 9:53 am

ziyuenlau wrote:If p is a positive integer, is p a prime number?

(1) p and p+1 have the same number of factors.
(2) pâˆ’1 is a factor of p.

Target question: Is p a prime number?

Given: p is a positive integer

Statement 1: p and p+1 have the same number of factors.
Let's TEST some values.
There are several values of p that satisfy statement 1. Here are two:
Case a: p = 2. This means that p+1 = 3. Notice that 2 and 3 both have the same number of factors (2 factors each). In this case, p IS prime
Case b: p = 14. This means that p+1 = 15. Notice that 14 and 15 both have the same number of factors (4 factors each). In this case, p is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: pâˆ’1 is a factor of p
Let's test some cases:
If p = 3, then p-1 = 2. Is 2 a factor of 3? No.
If p = 4, then p-1 = 3. Is 3 a factor of 4? No.
If p = 5, then p-1 = 4. Is 4 a factor of 5? No.
If p = 6, then p-1 = 5. Is 5 a factor of 6? No.
.
.
.
We can see that, if we keep going, p-1 will NEVER be a factor of p. Yet, statement 2 says that p-1 IS a factor of p.
Let's test the two positive integers that we haven't yet tested: 2 and 1
If p = 2, then p-1 = 1. Is 1 a factor of 2? YES! So, p COULD equal 2
If p = 1, then p-1 = 0. Is 0 a factor of 1? No
So, We can conclude that p MUST equal 2, which means p IS prime
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Fri May 19, 2017 10:20 am

Hi ziyuenlau,

This question can be solved by TESTing VALUES, but it will likely also require a bit of 'brute force' work. Sometimes the easiest/fastest way to get to the solution is to just put the pen on the pad and quickly list out the possibilities.

We're told that P is a POSITIVE INTEGER. We're asked if P is PRIME. This is a YES/NO question.

1) P and (P+1) have the same number of factors.

The information in this Fact might take a little bit of work to deal with, so let's brute force the possibilities until we find a couple of examples that match what we're told here:

P=1 .. 1 factor
P=2 .. 2 factors
P=3 .. 2 factors
P=4 .. 3 factors
P=5 .. 2 factors

P=6 .. 4 factors
P=7 .. 2 factors
P=8 .. 4 factors
P=9 .. 3 factors
P=10 .. 4 factors

P=11 .. 2 factors
P=12 .. 6 factors
P=13 .. 2 factors
P=14 .. 4 factors
P=15 .. 4 factors

We can now see two 'pairs' of numbers that have the SAME number of factors...
2 and 3; if P=2 then the answer to the question is YES
14 and 15; if P=14 then the answer to the question is NO
Fact 1 is INSUFFICIENT

2) (P-1) is a factor of P.

Again, let's start at P=1 and see what occurs...

P=1 .. 0 is not a factor of 1
P=2 .. 1 IS a factor of 2
P=3 .. 2 is not a factor of 3
P=4 .. 3 is not a factor of 4
Etc.

At this point, we can stop working - larger values of P will continue to yield the same result. The ONLY time that (P-1) is a factor of P is when P=2. Thus, there is ONLY one answer to the question (and it happens to be YES).
Fact 2 is SUFFICIENT

Final Answer: B

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

Contact Rich at [email protected]

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