FINDING LEAST COMMON MULTIPLES

The multiples of [beautiful math coming... please be patient] $\,2\,$ and $\,3\,$ are:

multiples of [beautiful math coming... please be patient] $\,2\,$:   $\,2\,$, $\,4\,$, $\,6\,$, $\,8\,$, $\,10\,$, $\,12\,$, $\,14\,$, $\,16\,$, $\,18\,$, $\,20\,$, $\,22\,$, $\,24\,$, etc.
multiples of [beautiful math coming... please be patient] $\,3\,$:   $\,3\,$, $\,6\,$, $\,9\,$, $\,12\,$, $\,15\,$, $\,18\,$, $\,21\,$, $\,24\,$, etc.

What numbers are multiples of both [beautiful math coming... please be patient] $\,2\,$ and $\,3\,$?
That is, what numbers appear in both lists above?
Answer: [beautiful math coming... please be patient] $\,6\,$, $\,12\,$, $\,18\,$, $\,24\,$, etc.
What is the least number that is a multiple of both [beautiful math coming... please be patient] $\,2\,$ and $\,3\,$?
Answer: [beautiful math coming... please be patient] $\,6\,$

The number [beautiful math coming... please be patient] $\,6\,$ is called the least common multiple of $\,2\,$ and $\,3\,$,
because it is a common multiple (i.e., it is a multiple of [beautiful math coming... please be patient] $\,2\,$ and a multiple of $\,3\,$),
and it is the smallest number with this property.

FINDING A LEAST COMMON MULTIPLE

If the individual numbers aren't too big,
then the following method of finding the least common multiple is often quick and easy:

Mentally go through the multiples of the largest number,
checking each of these multiples to see if all the other numbers go into it evenly.

For example, suppose you want the least common multiple of [beautiful math coming... please be patient] $\,3\,$, $\,5\,$, and $\,20\,$.
Of these three numbers, $\,20\,$ is the largest.
Go through the multiples of [beautiful math coming... please be patient] $\,20\,$,
stopping to check if each of these multiples is divisible by the other two numbers, [beautiful math coming... please be patient] $\,3\,$ and $\,5\,$:

Is [beautiful math coming... please be patient] $\,20\,$ divisible by both $\,3\,$ and $\,5\,$?   No;  it's not divisible by $\,3\,$.
Is [beautiful math coming... please be patient] $\,40\,$ divisible by both $\,3\,$ and $\,5\,$?   No;  it's not divisible by $\,3\,$.
Is [beautiful math coming... please be patient] $\,60\,$ divisible by both $\,3\,$ and $\,5\,$?   Yes!  So, $\,60\,$ is the least common multiple.

EXAMPLES:
The least common multiple of $\,2\,$ and $\,3\,$ is $\,\ldots\,$?
($\,3\,$ is the largest number. Go through its multiples:
Is $\,3\,$ divisible by $\,2\,$?  No.
Is $\,6\,$ divisible by $\,2\,$?  Yes.
Stop! $\,6\,$ is the least common multiple.)
The least common multiple of $\,4\,$ and $\,12\,$ is $\,\ldots\,$?
($\,12\,$ is the largest number. Go through its multiples:
Is $\,12\,$ divisible by $\,4\,$?  Yes.
Stop! $\,12\,$ is the least common multiple.)
The least common multiple of $\,6\,$ and $\,15\,$ is $\,\ldots\,$?
($\,15\,$ is the largest number. Go through its multiples:
Is $\,15\,$ divisible by $\,6\,$?  No.
Is $\,30\,$ divisible by $\,6\,$?  Yes.
Stop! $\,30\,$ is the least common multiple.)

More Efficient Methods for Finding the Least Common Multiple

Jo Johans calls the method presented above the ‘March of the Multiples’. I love it!

There are other methods that are better for finding the least common multiple,
when the numbers get bigger, or there are more of them:

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Renaming Fractions with a Specified Denominator

 
 
    
(an even number, please; max is 20)