Least Common Multiple

The Least Common Multiple (L.C.M.) of two numbers is the lowest number that can be divided both. It can be used to find the lowest common denominator when adding or subtracting fractions. Don't let the "least" in the name fool you - the LCM is no smaller than the largest of the numbers.

HINTS:
  LEAST - lowest or smallest.

COMMON - something shared or in common.

MULTIPLE - what you get when you multiply.

EXAMPLE:
Find the Least Common Multiple (L.C.M.) of 4 and 6.
 

Multiples of 4:   4, 8, 12, 16, 20, 24, 28, 32....
  1*4, 2*4, 3*4, 4*4, 5*4, 6*4, and so on

Multiples of 6:   6, 12, 18, 24, 30, 36, 42, 48....
  1*6, 2*6, 3*6, 4*6, 5*6, 6*6, and so on

Since 12, 24, 36, 48.... can be all divided by BOTH 4 and 6, each of them is a common multiple (and could be used as a common denominator of both fourths and sixths, for example). The lowest common multiple is 12, therefore it is the Least Common Multiple (L.C.M.) of 4 and 6.



You can also use the prime factorization method to find the Least Common Multiple:

EXAMPLE:
Find the Least Common Multiple (L.C.M.) of 10 and 12:

 

10 = 2 * 5 
and 
12 = 2 * 2 * 3
Count the maximum number of times each factor appears in either quantity. The product of those factors is the Least Common Multiple (L.C.M.):
L.C.M. = 2 * 2 * 3 * 5 = 60



Usually you can find the Least Common Multiple fairly easily by experimentation. Start with the larger number: Will the other number divide into it? If so, you have the L.C.M.; if not: What is the next multiple of the larger number -- is IT divisible by the other number? Continue until you find a number that is divisible by both numbers. If they share no common factors (other than one) then the Lowest Common Multiple will be the product of the two numbers.


For example the L.C.M. of 5 and 8 is 40
        since the only common factor is one,
        just multiply the numbers:   5*8 = 40.


Find the Lowest Common Multiple (L.C.M.) of 6 and 12.
        Will 6 go into 12? Yes.... L.C.M.= 12


Find the Lowest Common Multiple (L.C.M.) of 6 and 10.
        Will 6 go into 10?   No.... Next Multiple 2*10 = 20
        Will 6 go into 20?   No.... Next Multiple 3*10 = 30
        Will 6 go into 30?   Yes.... L.C.M. = 30



PRACTICE:     ("Click" on Check answer or use the answer key below.)

1)   Find the Least Common Multiple (L.C.M.) of 10 and 15.
  A) 15 B) 20 C) 30 D) 60 E) 150
 


2)   Find the Least Common Multiple (L.C.M.) of 4 and 9.
  A) 9 B) 18 C) 27 D) 36 E) 72
 


3)   Find the Least Common Multiple (L.C.M.) of 8 and 24.
  A) 24 B) 48 C) 72 D) 96 E) 192
 


4)   Find the Least Common Multiple (L.C.M.) of 5 and 12.
  A) 12 B) 30 C) 60 D) 90 E) 120
 


5)   Find the Least Common Multiple (L.C.M.) of 6 and 9.
  A) 9 B) 18 C) 36 D) 54 E) 90
 


6)   Find the Least Common Multiple (L.C.M.) of 4 and 10.
  A) 10 B) 20 C) 40 D) 80 E) 160
 


7)   Find the Least Common Multiple (L.C.M.) of 9 and 20.
  A) 20 B) 60 C) 90 D) 120 E) 180
 


8)   Find the Least Common Multiple (L.C.M.) of 7 and 35.
  A) 35 B) 70 C) 105 D) 245 E) 700
 


ANSWERS:   1) C     2) D     3) A     4) C     5) B     6) B     7) E     8) A



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