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is x an even integer?

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is x an even integer? Post Sat Jun 09, 2012 2:03 pm
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    If x is a positive number, is x an even integer?

    (1) 3x is an even integer.

    (2) 5x is an even integer.

    Can someone explain how to solve this?

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    eagleeye GMAT Destroyer!
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    Post Sat Jun 09, 2012 3:15 pm
    Hi PGMAT:

    The correct answer should be C. Let me explain:
    Remember that x is a positive number, not necessarily an integer.

    1) 3x is an even integer. Let 3x=2m (m is a positive integer). Here m=3/2*x
    If x=2/3, then 3*2/3=2 but x is not an integer.
    However if x=2, 3*2=6 is still an even ineger.
    Not sufficient.
    2) 5x is an even integer. Let 5x=2n (n is a positive integer). Here n=5/2*x
    by the same logic, x can be 2/5 or 2, which still makes it insufficient.

    EDIT: I had posted a harder to understand approach earlier, but I saw Mitch's post below and based the following explanation on that. I couldn't delete this post (the option dosen't come up for some reason), so I edited below, please thank Mitch if you like this approach as all credit goes to him)

    Now combining the two, 2*3x-5x =6x-5x = x.
    Since 6x, 5x are both even and we know that
    even - even = even
    Hence x is an even integer. Hence C is correct. Thank you Mitch!! Smile

    Let me know if this helps Smile



    Last edited by eagleeye on Sat Jun 09, 2012 5:42 pm; edited 4 times in total

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    Post Sat Jun 09, 2012 4:12 pm
    PGMAT wrote:
    If x is a positive number, is x an even integer?

    (1) 3x is an even integer.

    (2) 5x is an even integer.

    Can someone explain how to solve this?
    Statement 1: 3x is even.
    It's possible that x = 2, which is an even integer.
    It's possible that x = 2/3, which is not an even integer.
    INSUFFICIENT.

    Statement 2: 5x is even.
    It's possible that x = 2, which is an even integer.
    It's possible that x = 2/5, which is not an even integer.
    INSUFFICIENT.

    Statements 1 and 2 combined:
    5x-3x = even - even = even.
    Since 5x-3x = 2x, 2x is even.
    3x-2x = even - even = even.
    Since 3x-2x=x, x is even.
    SUFFICIENT.

    The correct answer is C.

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    Thanked by: eagleeye, PGMAT
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    IknowIcan Just gettin' started! Default Avatar
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    Post Sat Jun 09, 2012 7:36 pm
    Hi guys,

    Can you elaborate this part:

    3x-2x = even - even = even.
    Since 3x-2x=x, x is even.
    SUFFICIENT.


    Also, how can we prove C by using the sample numbers rather than numbers properties or algebra?


    Thanks!

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    Post Sat Jun 09, 2012 8:32 pm
    IknowIcan wrote:
    Hi guys,

    Can you elaborate this part:

    3x-2x = even - even = even.
    Since 3x-2x=x, x is even.
    SUFFICIENT.


    Also, how can we prove C by using the sample numbers rather than numbers properties or algebra?


    Thanks!
    When one even number is subtracted from another, the result is another even number.
    Thus:
    When even number 3x is subtracted from even number 5x, the result -- 2x -- is an another even number.
    When even number 2x is subtracted from even number 3x, the result -- x -- is another even number.
    Thus, when the two statements are combined, we know that x is even.
    SUFFICIENT.

    If we stick to plugging in values:
    Non-integer values that satisfy statement 1 are 2/3, 4/3, 8/3, etc.
    None of these values satisfies statement 2.
    Non-integer values that satisfy statement 2 are 2/5, 4/5, 6/5, 8/5, etc.
    None of these values satisfies statement 1.
    Thus, the only values that satisfy BOTH statements are EVEN INTEGERS:
    2, 4, 6, 8, etc.
    Thus, to satisfy both statements, x must be even.
    SUFFICIENT.

    Generally, plugging in numbers is the easiest way to prove that a statement is INSUFFICIENT.
    But to prove that a statement is SUFFICIENT, algebra or some other sort of reasoning often is more efficient.

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    dhonu121 GMAT Destroyer! Default Avatar
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    Post Mon Jun 11, 2012 2:03 am
    Quote:
    If x is a positive number, is x an even integer?

    (1) 3x is an even integer.

    (2) 5x is an even integer.

    Can someone explain how to solve this?
    1. 3x is even.
    so 3x=2 or x=3/2=1.5. Not even.
    also 3x=4 or x = 4/3=1.33. Not Even.
    also 3x=6 or x=2. Hence Even.
    Thus for some cases its even for some its odd. Hence nothing can be said from this info.
    Hence Insufficient.

    2. 5x is even.
    Same as above.
    5x=2 or 4 or 6. x is not even.
    5x=10 or 20 or 30. X is even.
    Thus nothing can be said.
    Hence Insufficient.

    Combining 1 and 2.
    3x is even and 5x is even.
    If 3x=2 or 4 or 8, then from that value of x, 5x is not even.
    If 3x=6,12,18 then from that value of x, 5x is also even.
    Thus for 1 and 2 to be true, x has to be even.
    Hence C is the answer.

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