x!+x+1

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j_shreyans Legendary Member Default Avatar
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x!+x+1

Post Mon Sep 08, 2014 9:56 pm
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  • Lap #[LAPCOUNT] ([LAPTIME])
    If x is a positive integer greater than 1, is x! + x + 1 a prime number?

    (1) x < 10

    (2) x is odd

    OAB

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    Post Mon Sep 08, 2014 11:36 pm
    HI j_shreyans,

    This question is perfect for TESTing VALUES.

    We're told that X is a POSITIVE INTEGER greater than 1. We're asked if X! + X + 1 is prime. This is YES/NO question.

    Fact 1: X < 10

    If X = 2, then 2! + 2 + 1 = 5 and the answer to the question is YES.
    If X = 3, then 3! + 3 + 1 = 10 and the answer to the question is NO.
    Fact 1 is INSUFFICIENT

    Fact 2: X is odd

    If X =3, then 3! + 3 + 1 = 10 and the answer to the question is NO.
    If X = 5, then 5! + 5 + 1 = 126 and the answer to the question is NO.

    From here, it's worthwhile to discuss a Number Property that applies….

    Since X is Odd AND greater than1, X! will be EVEN (since the product includes at least 1 even number). Adding X (which is odd) + 1 (which is also odd), the sum MUST BE EVEN. The sum will be divisible by 2, so it will NOT be prime and the answer to the question will ALWAYS BE NO.
    Fact 2 is SUFFICIENT

    Final Answer: B

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

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    Post Wed Sep 10, 2014 10:51 am
    j_shreyans wrote:
    If x is a positive integer greater than 1, is x! + x + 1 a prime number?

    (1) x < 10
    (2) x is odd
    Target question: Is (x! + x + 1) a prime number?

    Given: x is a positive integer greater than 1

    Statement 1: x < 10
    There are several values of x that satisfy this condition. Here are two:
    Case a: x = 2, in which case x! + x + 1 = 2! + 2 + 1 = 5. So, YES, x! + x + 1 IS prime
    Case b: x = 3, in which case x! + x + 1 = 3! + 3 + 1 = 10. So, NOT, x! + x + 1 is NOT prime
    Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

    Statement 2: x is ODD
    IMPORTANT: if x is an integer greater than 1, then x! must be EVEN
    Notice that:
    2! = (2)(1) = 2 (EVEN)
    3! = (3)(2)(1) = 6 (EVEN)
    4! = (4)(3)(2)(1) = 24 (EVEN)
    5! = (5)(4)(3)(2)(1) = 120 (EVEN)
    etc
    -------
    So, if x is ODD, then x! + x + 1 = EVEN + ODD + 1
    = EVEN
    If x! + x + 1 is even (and greater than 2), then x! + x + 1 definitely CANNOT be prime
    Since we can answer the target question with certainty, statement 2 is SUFFICIENT

    Answer = B

    Cheers,
    Brent

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