Pre-Algebra: LCM

1.
$$4,6$$
Solution
2.
$$8,12$$
Solution
3.
$$14,7,28$$
Solution
4.
$$3,4,8$$
Solution
5.
$$2,6,28$$
Solution
6.
$$13,2$$
Solution

Lowest Common Multiple - Introduction

A multiple of a number is the result of multipling that number by an integer (whole number). Consider the multiples of
$$3$$
and
$$4$$
:

Multiples of $$3$$

$$3\times 1=3$$

$$3\times 2=6$$

$$3\times 3=9$$

$$3\times 4=12$$

$$3\times 5=15$$

$$3\times 6=18$$

$$3\times 7=21$$

$$3\times 8=24$$

Multiples of $$4$$

$$4\times 1=4$$

$$4\times 2=8$$

$$4\times 3=12$$

$$4\times 4=16$$

$$4\times 5=20$$

$$4\times 6=24$$

$$4\times 7=28$$

$$4\times 8=32$$

Notice that there are certain multiples that are the same for both
$$3$$
and
$$4$$
, since both can multiply to
$$12$$
and
$$24$$
. For many math problems, it’s helpful to know the smallest multiple that a set of numbers share. This is called the least common multiple, or lowest common multiple, also written as LCM. In this case, the least common multiple of
$$3$$
and
$$4$$
is
$$12$$
.

Methods of Finding the Lowest Common Multiple

There are two methods to help you easily find the LCM.

Method 1: Listing Multiples

This is the method we used above. Take two (or more) numbers and list their multiple lists until you find a common answer. Consider the lists of
$$4$$
,
$$7$$
and
$$8$$
.

Multiples of $$4$$

$$4\times 1=4$$

$$4\times 2=8$$

$$4\times 3=12$$

$$4\times 4=16$$

$$4\times 5=20$$

$$4\times 6=24$$

$$4\times 7=28$$

$$4\times 8=32$$

$$4\times 9=36$$

$$4\times 10=40$$

$$4\times 11=44$$

$$4\times 12=48$$

$$4\times 13=52$$

Multiples of $$7$$

$$7\times 1=7$$

$$7\times 2=14$$

$$7\times 3=21$$

$$7\times 4=28$$

$$7\times 5=35$$

$$7\times 6=42$$

$$7\times 7=49$$

$$7\times 8=56$$

$$7\times 9=63$$

$$7\times 10=70$$

$$7\times 11=77$$

$$7\times 12=84$$

Multiples of $$8$$

$$8\times 1=8$$

$$8\times 2=16$$

$$8\times 3=24$$

$$8\times 4=32$$

$$8\times 5=40$$

$$8\times 6=48$$

$$8\times 7=56$$

$$8\times 8=64$$

$$8\times 9=72$$

$$8\times 10=80$$

$$8\times 11=88$$

$$8\times 12=96$$

While
$$4$$
and
$$8$$
share multiples like
$$16$$
,
$$24$$
,
$$32$$
and
$$20$$
, the lowest common multiple for all three numbers is
$$56$$
.

Method 2: Prime Factors

Start by listing the prime factors of each number. Prime factors are the set of prime numbers that, when multiplied, give the original number. Let’s use this method to find the LCM of
$$20$$
and
$$25$$
.
Prime factors of
$$20$$
$$5\times 2\times 2$$
Prime factors of
$$25$$
$$5\times 5$$
Next, multiply each factor the most number of times it appears in either prime factorization. This means we need to multiply
$$2\times 2\times 5\times 5$$
because
$$2$$
occurs twice in one set of prime factors and
$$5$$
occurs twice in the other. The remaining
$$5$$
in the first set of factors does not need to be multiplied. This gives us
$$100$$
, which is the lowest common multiple for both
$$20$$
and
$$25$$
.

What's Next

Ready to give it a try? Use either method to calculate the LCM in our practice problems at the top of this page.
Our LCM problems are designed to help students quickly grasp and master the skill of discovering least common multiples for any set of numbers. You can also acquire skills in other topics via our practice problems. We hope that through these problems and solutions, you will be empowered with the math concepts that you need to succeed.