Finding Least Common 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.)

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? Read the text.

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 Specific Denominator

(an even number, please)