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What is the LCM of x, 6 and 9

This topic has 3 expert replies and 2 member replies
melguy Master | Next Rank: 500 Posts
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What is the LCM of x, 6 and 9

Post Sat Jun 15, 2013 10:16 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Hello All

    Please help me with the problem. This has been discussed earlier in the forum but I did not understand the explanations.

    Also, I am curious to know if there is any shortcut for this problem.

    Thanks


    * If x is a positive integer, what is the LCM of x,6 and 9?
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    Post Sun Jun 16, 2013 6:30 am
    Quote:
    If x is a positive integer, what is the least common multiple of x, 6, and 9 ?
    (1) The least common multiple of x and 6 is 30.
    (2) The least common multiple of x and 9 is 45.
    Target question: What is the LCM of x, 6 and 9?

    I'll show you two different approaches.
    This first approach uses requires us to be able to think of pairs of values that have given LCM's.
    This is a useful skill to have on the GMAT.

    Statement 1: The least common multiple of x and 6 is 30.
    So, what are some possible values of x?
    If the LCM of x and 6 is 30, then x could equal 5, 10, 15 or 30
    Let's check each possible value of x.
    - If x = 5, then the LCM of x, 6, and 9 is 90
    - If x = 10, then the LCM of x, 6, and 9 is 90
    - If x = 15, then the LCM of x, 6, and 9 is 90
    - If x = 30, then the LCM of x, 6, and 9 is 90
    So, even though x can have several different values, it must be the case that the LCM of x, 6, and 9 is 90
    Since we can answer the target question with certainty, statement 1 is SUFFICIENT

    Statement 2: The least common multiple of x and 9 is 45.
    So, what are some possible values of x?
    If the LCM of x and 6 is 30, then x could equal 5, 15 or 45
    Let's check each possible value of x.
    - If x = 5, then the LCM of x, 6, and 9 is 90
    - If x = 15, then the LCM of x, 6, and 9 is 90
    - If x = 45, then the LCM of x, 6, and 9 is 90
    So, even though x can have several different values, it must be the case that the LCM of x, 6, and 9 is 90
    Since we can answer the target question with certainty, statement 2 is SUFFICIENT

    Answer = D

    Cheers,
    Brent

    _________________
    Brent Hanneson – Founder of GMATPrepNow.com
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    melguy Master | Next Rank: 500 Posts
    Joined
    21 Mar 2011
    Posted:
    335 messages
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    2 members
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    37 times
    Target GMAT Score:
    650
    Post Sun Jun 16, 2013 6:39 am
    Awesome explanation! Thanks a lot for your help Brent!

    Also, just a compliment - your GMAT PrepNow videos are excellent Smile

    Post Sun Jun 16, 2013 6:50 am
    Quote:
    If x is a positive integer, what is the least common multiple of x, 6, and 9 ?
    (1) The least common multiple of x and 6 is 30.
    (2) The least common multiple of x and 9 is 45.
    Target question: What is the LCM of x, 6 and 9?

    ASIDE: The LCM tells us about the prime factors that numbers have in common.
    For example: The LCM of 20 and 12 is 60
    60 = (2)(2)(3)(5). So, the prime factorization of 60 has two 2's, one 3, and one 5.

    Now examine the prime factorizations of 20 and 12
    20 = (2)(2)(5)
    12 = (2)(2)(3)
    Notice that each prime factorization has no more than two 2's, one 3, and one 5 in it.
    Also notice that the combined prime factorizations of 20 and 12 account for the two 2's, one 3, and one 5 that we find in the prime factorization of 60.

    Statement 1: The least common multiple of x and 6 is 30
    30 = (2)(3)(5)
    6 = (2)(3), so we've already accounted for the one 2 and one 3 in the prime factorization of 30
    We're missing only a 5
    So, the prime factorization of x must have a 5 in it.
    The prime factorization of x could also have a 2 or 3 in it, but they aren't required.
    So, the possible values of x are 5, 10 (aka 5 times 2), 15 (aka 5 times 3) and 30 (aka 5 times 2 times 3)
    As we saw in my earlier post, if we consider all of these possible values of x, the LCM of x, 6 and 9 is always 90
    Since we can answer the target question with certainty, statement 1 is SUFFICIENT

    Statement 2: The least common multiple of x and 9 is 45.
    45 = (3)(3)(5)
    9 = (3)(3), so we've already accounted for the two 3's in the prime factorization of 45
    We're missing only a 5
    Using the same logic as above, the possible values of x are 5, 15 and 45
    If we consider all of these possible values of x, the LCM of x, 6 and 9 is always 90
    Since we can answer the target question with certainty, statement 2 is SUFFICIENT

    Answer = D

    Cheers,
    Brent

    _________________
    Brent Hanneson – Founder of GMATPrepNow.com
    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: melguy, Anaira Mitch
    GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!
    Post Sun Jun 16, 2013 6:51 am
    melguy wrote:
    Awesome explanation! Thanks a lot for your help Brent!

    Also, just a compliment - your GMAT PrepNow videos are excellent Smile
    Thanks melguy.

    Cheers,
    Brent

    _________________
    Brent Hanneson – Founder of GMATPrepNow.com
    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

    GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!
    Post Fri Jun 21, 2013 6:52 am
    lcm of x and 6 is 30...other multiples of x and 6 are 30x2=60, 30x3=90, 30x4=120....so among these values 90 is a multiple of 9. so lcm of x,6,9 is 90.

    similarly..
    lcm of x and 9 is 45.. other multiples of x and 9 are 45x2=90, 45x3=135.... so among these 90 is a multiple of 6.... so lcm of x,6,9 is 90...
    so both the statements are sufficient individually.

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