Finding the Least Common Multiple

You probably learned about least common multiples in the context of adding and subtracting fractions with different denominators. The least common multiple of the two denominators would be the same as the least common denominator. For example, if you wanted to add , you could use a common denominator of 12 times 8, or 96. This would result in which equals . Did you immediately recognize this as equal to ? If not, try using 24 as the common denominator. = . No simplification was necessary! Why did we use 24 as the common denominator? It happens to be the least common multiple (LCM) of 8 and 12. How do we find it? We’ll see in Example 1 below.

What is a least common multiple?

Let x and y be two nonzero whole numbers. The least common multiple of x and y is the smallest whole number that is a multiple of both x and y. This is written as lcm (x, y) or LCM (x, y).

Here is an alternative definition: The least common multiple of x and y is the smallest number that has both x and y as factors.

For example:

                                    a) lcm (24, 36) = 72

                                    b) LCM (12, 3) = 12

                                    c) LCM (18, 4) = 36

If you have more than two nonzero whole numbers, then the least common multiple of all the numbers is the smallest whole number that is a multiple of each number.

Example 1

Find LCM (8, 12)

Solution 1

Step 1: Find the prime factorization of 8 and 12. Use factor trees or any method you prefer.

8 = 2³ 12 = 2² • 3

Step 2: The LCM is the product of the greatest power of every prime in the factorizations of the numbers. In this case, LCM (8, 12) = 2³ • 3 = 24.

Example 2

Find LCM(60, 40)

Solution 2

Step 1: Find the prime factorization of 60 and 40.

Step 2: The LCM is the product of the greatest power of every prime in the two
factorizations. In this case,

Example 3

Find lcm(18, 30, 54).

Solution 3

Step 1: Find the prime factorization of 18, 30 and 54

18 = 2 • 3² 30 = 2 • 3 • 5 54 = 2 • 3³

Step 2:

Note that        270 = 18 • 15
                        270 = 30 • 9
                        270 = 54 • 5

and that the numbers 15, 9, and 5 have no common factors. This proves that 270 is the least common multiple of 18, 30, and 54.

In algebra, we use this concept in more sophisticated situations involving variables, particularly in algebraic fractions. Least common denominators greatly simplify our work in algebra, so being able to use them with numbers first is very helpful!