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If p is positive, is p prime?

This topic has 2 expert replies and 0 member replies

If p is positive, is p prime?

Post Thu Nov 02, 2017 8:51 am
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    If p is positive, is p prime?

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

    What's the best way to determine which statement is sufficient?

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    Post Thu Nov 09, 2017 10:12 pm
    ardz24 wrote:
    If p is positive, is p prime?

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

    What's the best way to determine which statement is sufficient?
    (1) p^3 has exactly 4 distinct factors.

    Case 1: Assume that p is prime, thus, the number of factors of p^3 are 1, p, p^2, and p^3. There are four factors.
    Case 2: Assume that p is non-prime; let's assume = 4, a convenient value. The number of factors of 4^3 (= 64) are 1, 2, 4, 8, 16, 32, and 64. The number of factors is not four, thus, this case is invalid.

    In case, you wish to try Case 1 with some values such as 2 and 101, you may. For p = 2, the number 2^3 has 1, 2, 4, and 8 as factors (four); for p = 10, the number 101^3 has 1, 101, 101^2, and 101^3 as factors (four).

    Sufficient.

    (2) p^2 - p - 6 = 0

    => p^2 - 3p + 2p - 6 = 0 => p(p - 3) + 2(p - 3) = 0 => p = 3 or -2. Since p is positive, p = 3, a prime number. Sufficient.

    The correct answer: D

    Hope this helps!

    -Jay
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    Post Wed Nov 15, 2017 9:08 pm
    Hi ardz24,

    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.

    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

    _________________
    Contact Rich at Rich.C@empowergmat.com

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