SIMPLIFYING EXPRESSIONS:  LEAST COMMON MULTIPLE OF TWO MONOMIALS

 

The least common multiple is the smallest expression that the two monomials will divide into.  Even though the term is “least” the expression has to be as big as the biggest term you are considering.  Finding a Least Common Multiple is the same as finding a Lowest Common Denominator.

 

ALEKS shows you the factor method for finding the least common multiple of two numbers so I will show you an alternate method.  You can simply take multiples of the largest number until you find one that the other number(s) go into.

 

Example:  Find the least common multiple for 15 and 6.  Start with multiples of 15 since it is the larger number.  Since 6 does not divide evenly into 15 we go to the next multiple of 15 which is 2 times 15 or 30.  This is the number we want because 6 divides evenly into 30 (5 times in fact).

 

The other method is to factor the numbers and use each different factor in the Least Common Multiple.

Our example with 15 and 6 would work as follows:

 

15 = 3 x 5   and 6 = 2 x 3

 

The least common multiple must have factors of 2, 3, and 5.  Notice that 2 x 3 x 5 = 30…the same answer as before.  Note that we used 3 only once because it is only in each denominator once.

 

If a number has the same factor more than once, we need that factor in the least common multiple the most number of times it appears in any ONE of the numbers.

 

Example:

Find the least common multiple of 12 and 27.

 

12 factors into 2 x 2 x 3             and 27 factors into 3 x 3 x 3

 

The least common multiple will need 2 factors of 2 and 3 factors of 3 because 2 appears twice in 12 and 3 appears 3 times as a factor of 27.  The least common multiple is 2 x 2 x 3 x 3 x 3 = 108.

 

Using the other method, we would have to keep finding multiples of 27 until we find one that 12 divides evenly into:

12 does not divide evenly into the first two multiples 27 and 54, but it does divide into 108 (9 times).

 

 

Now let’s throw variables into the mix.  Since they don’t need to be factored, we simply need to use each different variable that appears in any of the numbers to the highest power that is used.

 

Example: Find the least common multiple of

 

Using one of the techniques from above, we find the least common multiple of 16 and 12 to be 48.  The variables in blue are the ones to the highest powers.  Our least common multiple for these two monomials is