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Math Help - Lcm

  1. #1
    Newbie Steves5a's Avatar
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    Lcm

    how would you work the following to get the lowest common multiple:


    16
    40

    I think that it is 2 but not sure
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  2. #2
    Super Member Rebesques's Avatar
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    e?

    I think you mixed least common multiples and greatest common divisors. The multiples are always number greater than the two given.


    There is a neat way for working out least common multiples and greatest common divisors: Prime factorization. Take 16 and 40, and express them as products of numbers, non divisible by others: 16=2*8=2^4 (power) , and 40=5*8=5*2^3. For the least common multiple [16,40], we have that products common in both places get out of the brackets:

    [16,40]=[2^4,5*2^3]=2^3*[2,5] (2^3 is common in both places)
    =8*[2,5]=8*2*5=80
    ( 'cause 2 and 5 have no common divisors, so bracket becomes product)


    For the greatest common divisor, (16,40) the same considerations hold, accept that when two numbers in the parentheses have no common divisors, the result is 1 .We have then

    (16,40)=(2^4,5*2^3)=2^3*(2,5)=8*1=8.
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  3. #3
    MHF Contributor
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    Here is one way.

    LCM of set of numbers is the smallest number that can be divided evenly by any of the number in the set.

    For 16 and 40, their LCM is the smallest number that when divided by 16 or by 40, the quotients are whole numbers only.

    If we get the prime factors of 16 and 40, the product (multiplication) of the primes in their highest degree is the LCM.

    Prime-factorization of 16
    = 2,2,2,2
    = 2^4
    >>>2 is the only prime, and its degree is 4.

    Prime-factorization of 40
    = 2,2,2,5
    = 2^3,5^1
    >>>2 and 5 are the primes.
    >>>2 has a degree of 3 only, lower than the 4 in 16, so we will use the 2^4.
    >>>5 has a degree of 1 and it is the highest degree of 5, so we use 5^1.

    Hence,
    LCM = (2^4)*(5^1) = 16*5 = 80 --------answer.

    Check,
    80/16 = 5 ----whole number.
    80/40 = 2 ----whole number too.
    Therefore, okay.
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