Finding Least Common Multiples

Need some basic practice with multiples first? Practice with Multiples

The multiples of $\,2\,$ and $\,3\,$ are:

multiples of $\,2\,$:   $\,2\,$, $\,4\,$, $\,6\,$, $\,8\,$, $\,10\,$, $\,12\,$, $\,14\,$, $\,16\,$, $\,18\,$, $\,20\,$, $\,22\,$, $\,24\,$, etc.

multiples of $\,3\,$:   $\,3\,$, $\,6\,$, $\,9\,$, $\,12\,$, $\,15\,$, $\,18\,$, $\,21\,$, $\,24\,$, etc.

What numbers are multiples of both $\,2\,$ and $\,3\,$? That is, what numbers appear in both lists above?

Answer: $\,6\,$, $\,12\,$, $\,18\,$, $\,24\,$, etc.

What is the least number that is a multiple of both $\,2\,$ and $\,3\,$?

Answer: $\,6\,$

The number $\,6\,$ is called the least common multiple of $\,2\,$ and $\,3\,$, because it is a common multiple (i.e., it is a multiple of $\,2\,$ 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 $\,3\,$, $\,5\,$, and $\,20\,$. Of these three numbers, $\,20\,$ is the largest. Go through the multiples of $\,20\,$, stopping to check if each of these multiples is divisible by the other two numbers, $\,3\,$ and $\,5\,$:

Is $\,20\,$ divisible by both $\,3\,$ and $\,5\,$?   No;  it's not divisible by $\,3\,$.

Is $\,40\,$ divisible by both $\,3\,$ and $\,5\,$?   No;  it's not divisible by $\,3\,$.

Is $\,60\,$ 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 Johansen 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:

• One method is the prime factorization method . Interested?
• Zip to WolframAlpha and type in (say):

least common multiple of 18, 36, 90
or
lcm(18,36,90)

Practice

 The least common multiple of and is: