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

This topic has 2 expert replies and 1 member reply
hazelnut01 Master | Next Rank: 500 Posts Default Avatar
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If p is a positive integer, is p a prime number? (1) p and p

Post Fri May 19, 2017 6:16 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    If p is a positive integer, is p a prime number?

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

    OA=B

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    susheelh Newbie | Next Rank: 10 Posts Default Avatar
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    Post 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: p-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) p-1 is a factor of p.

    OA=B

    Post 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) p-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: p-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

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    Post 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

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