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MGMAT smallest prime factor - Need expert help

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voodoo_child GMAT Destroyer! Default Avatar
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MGMAT smallest prime factor - Need expert help Post Mon May 07, 2012 9:37 am
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  • Lap #[LAPCOUNT] ([LAPTIME])
    X = (3 * 5 * 7 ....47) {all +ve odd integers}

    Y= X + 2

    what's the smallest prime factor of Y?

    I know that it will be greater than 47. I am not sure how to calculate 53.

    Any help?

    OA = 53

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    voodoo_child GMAT Destroyer! Default Avatar
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    Post Mon May 07, 2012 2:40 pm
    Can any of the experts please help me? Thanks

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    Bill@VeritasPrep GMAT Instructor
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    Post Mon May 07, 2012 2:58 pm
    Where is the question from, and what were the answer choices?

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    Post Mon May 07, 2012 3:18 pm
    Sure, Bill - The question is from Manhattan GMAT.
    Answer choices were -

    1) Y>50
    2) 30 <= y <= 50
    3) 10 <= Y <= 30
    4) 3<= y <10
    5) y=2

    MGMAT says that y = 53 is the minimum possible prime factor of (3...47)+2

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    Post Mon May 07, 2012 3:23 pm
    The trick here is to realize that any of the prime numbers that fall between 3 and 47 (inclusive) are already factors of 3#47. As a result, they cannot also be factors of 3#47 + 2. 48, 49, and 50 are not primes, so y must be greater than 50.

    The trap answer here is 2, I believe. It's conspicuously absent from 3#47, so it could be tempting. Since # represents the multiplication of odd integers only, we know that 3#47 = 3*5*7*...*49, which will be an odd product. An odd plus an even gives us an odd sum, so (3#47) + 2 is odd. Therefore, 2 cannot be a factor.

    I've seen variations on this question; one said that x was the product of all even integers from 2 to 50, then asked for the smallest prime factor of x + 1.

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    Post Mon May 07, 2012 3:35 pm
    Bill@VeritasPrep wrote:
    The trick here is to realize that any of the prime numbers that fall between 3 and 47 (inclusive) are already factors of 3#47. As a result, they cannot also be factors of 3#47 + 2. 48, 49, and 50 are not primes, so y must be greater than 50.

    The trap answer here is 2, I believe. It's conspicuously absent from 3#47, so it could be tempting. Since # represents the multiplication of odd integers only, we know that 3#47 = 3*5*7*...*49, which will be an odd product. An odd plus an even gives us an odd sum, so (3#47) + 2 is odd. Therefore, 2 cannot be a factor.

    I've seen variations on this question; one said that x was the product of all even integers from 2 to 50, then asked for the smallest prime factor of x + 1.
    Sure, Bill. I got this one correct. But why does OE say that minimum prime factor = 53?

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    Post Mon May 07, 2012 3:42 pm
    Well, the smallest prime number greater than 50 is 53.

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    Post Mon May 07, 2012 3:59 pm
    Bill@VeritasPrep wrote:
    Well, the smallest prime number greater than 50 is 53.
    ok. But how do I know whether it's a factor of y? Can you please help me?

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    Post Mon May 07, 2012 4:05 pm
    Doesn't the explanation say that 53 is the minimum possible prime factor?

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    dabral Rising GMAT Star
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    Post Mon May 07, 2012 10:57 pm
    What Bill is saying is that we can only conclude that the smallest prime factor of y must be greater than 47, for all we know it could be 53(least prime number greater than 47), or it could be 89, or Y itself could be prime. We have no way of knowing that based on the expression given for Y.

    This question is based on a difficult GMATPrep question that Bill alluded to, Here is the link:
    http://www.beatthegmat.com/functions-t85150.html#377237

    Another problem similar in concept but a little bit easier is Problem Solving#77 from Official Guide GMAT 13th Edition.

    Cheers,

    Dabral

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    Post Tue May 08, 2012 2:26 am
    It's a good problem. A simple logic breaks the puzzle out.
    consider a small example here:
    3*11 + 2 = 35 is divisible by 7,but 3*7 + 2 = 23 is not. What can be inferred from this?
    First case: (3*11 + 2)
    The two parts 3*11 and 2 are not divisible by 7 individually, but the sum does.
    Second case: (3*7 + 2)
    As 3*7 is already a factor of 7, the added value must also be divisible by 7, for the final sum is to be divisible by 7. But the added value 2 is not divisible by 7 and hence the whole part does not.
    The basic funda here is - "if X is divisible by p and Y is divisible by p then, X+Y is also divisble by p".

    Now, come back to our problem.
    (3*5 + 2) is not divisible by 3 and 5 as the second part '2' is not divisible by them.
    similarly (3*5*7 + 2) is not divisible by any of 3,5, and 7.
    For (3*5*7 + x) is to be divisible by 3, x must be 3 or multiple of 3. Similar logic is applicable for the divisibility by 5 and 7.

    As '2' is a number which is not divisible by any of the odd numbers 3,5,7,...47,
    the number (3*5*7*...47)+2 is not divisible by any of 3,5,7,...47.
    And so the minimum possible prime number to be considered for divisiblity is 53.

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