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If 3 < x < 100, for how many values of x is x/3 the sq

This topic has 3 expert replies and 2 member replies

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rsarashi Master | Next Rank: 500 Posts Default Avatar
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If 3 < x < 100, for how many values of x is x/3 the sq

Post Sat Apr 29, 2017 8:45 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    If 3 < x < 100, for how many values of x is x/3 the square of a prime number?

    (A) Two
    (B) Three
    (C) Four
    (D) Five
    (E) Nine

    OAB

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    Post Sat Apr 29, 2017 11:58 pm
    rsarashi wrote:
    If 3 < x < 100, for how many values of x is x/3 the square of a prime number?

    (A) Two
    (B) Three
    (C) Four
    (D) Five
    (E) Nine

    OAB
    We have a situation: x/3 the square of a prime number such that 3 < x < 100.

    3 < x < 100 => 1 < x/3 < 100/3 => 1 < x/3 < 33.33 => 1 < p^2 < 33.33; where p is a prime number

    Let us list down few prime numbers; they are 2, 3, 5, 7, 11, etc.

    Since x/3 is a square of a prime number, only prime numbers that would qualify are 2, 3, & 5.

    The correct answer: B

    Hope this helps!

    Relevant book: Manhattan Review GMAT Number Properties Guide

    -Jay
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    Thanked by: mbaclaus86, rsarashi
    Post Sun Apr 30, 2017 7:22 am
    rsarashi wrote:
    If 3 < x < 100, for how many values of x is x/3 the square of a prime number?

    (A) Two
    (B) Three
    (C) Four
    (D) Five
    (E) Nine

    OAB
    We want values of x (where 3 < x < 100) such that x/3 is the square of a prime number.
    So, let's start checking squares of prime numbers.
    Some prime numbers are 2, 3, 5, 7, 11, etc

    2² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
    3² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
    5² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
    7² = 49 and (3)(49) = 147. No good. We need values of x such that 3 < x < 100

    So, there are exactly 3 values of x that meet the given conditions.
    Answer: B

    Cheers,
    Brent

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    Thanked by: mbaclaus86, rsarashi
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    Top Member

    rsarashi Master | Next Rank: 500 Posts Default Avatar
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    Post Wed May 03, 2017 9:19 am
    Quote:
    2² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
    3² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
    5² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
    7² = 49 and (3)(49) = 147. No good. We need values of x such that 3 < x < 100
    Hi Brent ,

    Thank you so much for your reply.

    Just a quick question can you please advise that why did you multiply with 3? We have to divide x/3.

    Please explain sir.

    Post Wed May 03, 2017 9:27 am
    rsarashi wrote:
    Quote:
    2² = 4 and (3)(4) = 12. So, x = 12 meets the given conditions.
    3² = 9 and (3)(9) = 27. So, x = 27 meets the given condition12
    5² = 25 and (3)(25) = 75. So, x = 75 meets the given conditions.
    7² = 49 and (3)(49) = 147. No good. We need values of x such that 3 < x < 100
    Hi Brent ,

    Thank you so much for your reply.

    Just a quick question can you please advise that why did you multiply with 3? We have to divide x/3.

    Please explain sir.
    Great question.

    First off, we want x/3 to be the square of a prime number.
    Since all prime numbers are integers, the square of a prime number will be an integer.

    In order for x/3 to be an integer, it must be the case that x is divisible by 3.
    Another way to put it is that x must be a multiple of 3.
    This why I took each squared value and multiplied it by 3.

    Cheers,
    Brent

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    Use our video course along with Beat The GMAT's free 60-Day Study Guide

    Check out the online reviews of our course
    Come see all of our free resources

    Thanked by: rsarashi
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    Top Member

    rsarashi Master | Next Rank: 500 Posts Default Avatar
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    Post Sat May 06, 2017 6:46 pm
    Quote:
    Great question.

    First off, we want x/3 to be the square of a prime number.
    Since all prime numbers are integers, the square of a prime number will be an integer.

    In order for x/3 to be an integer, it must be the case that x is divisible by 3.
    Another way to put it is that x must be a multiple of 3.
    This why I took each squared value and multiplied it by 3.
    Hi Brent ,

    Perfect! All clear.

    Thanks

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