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What is the LCM of x, 6 and 9

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melguy Really wants to Beat The GMAT! Default Avatar
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What is the LCM of x, 6 and 9 Post Sat Jun 15, 2013 10:16 pm
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    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

    _________________
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    melguy Really wants to Beat The GMAT! Default Avatar
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    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 - Creator of GMAT Prep Now
    - Check out GMAT Prep Now’s online course at http://www.gmatprepnow.com/
    - Use our $139 video course along with Beat The GMAT's free 60-Day Study Guide
    - Watch hours of free videos on DS, RC and AWA
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    Thanked by: melguy
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    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 - Creator of GMAT Prep Now
    - Check out GMAT Prep Now’s online course at http://www.gmatprepnow.com/
    - Use our $139 video course along with Beat The GMAT's free 60-Day Study Guide
    - Watch hours of free videos on DS, RC and AWA
    - NEW: 5 full-length practice tests with every course

    Study Smart! Use Beat The GMAT’s FREE 60-Day Study Guide in conjunction with GMAT Prep Now’s video course and reach your target score in 2 months! With two money-back guarantees, you can try us out risk-free.
    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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