The Least Common Multiple

Read this text for a description of multiples and the method to determine the least common multiple of a given set of numbers. Complete the practice questions and check your answers.

Multiples

When a whole number is multiplied by other whole numbers, with the exception of zero, the resulting products are called multiples of the given whole number. Note that any whole number is a multiple of itself.

Sample Set A

Multiples of 2
Multiples of 3 Multiples of 8 Multiples of 10
$2×1=2$
$3×1=3$ $8×1=8$ $10×1=10$
$2×2=4$
$3×2=6$ $8×2=16$ $10×2=20$
$2×3=6$
$3×3=9$ $8×3=24$ $10×3=30$
$2×4=8$
$3×4=12$ $8×4=32$ $10×4=40$
$2×5=10$
$3×5=15$ $8×5=40$ $10×5=50$

Common Multiples

There will be times when we are given two or more whole numbers and we will need to know if there are any multiples that are common to each of them. If there are, we will need to know what they are. For example, some of the multiples that are common to 2 and 3 are 6, 12, and 18.

Sample Set B

We can visualize common multiples using the number line.

Notice that the common multiples can be divided by both whole numbers.

The Least Common Multiple (LCM)

Notice that in our number line visualization of common multiples (above), the first common multiple is also the smallest, or least common multiple, abbreviated by LCM.

Least Common Multiple

The least common multiple, LCM, of two or more whole numbers is the smallest whole number that each of the given numbers will divide into without a remainder.

The least common multiple will be extremely useful in working with fractions.

Finding the Least Common Multiple

Finding the LCMTo find the LCM of two or more numbers:

• Write the prime factorization of each number, using exponents on repeated factors.
• Write each base that appears in each of the prime factorizations.
• To each base, attach the largest exponent that appears on it in the prime factorizations.
• The LCM is the product of the numbers found in step 3.

There are some major differences between using the processes for obtaining the GCF and the LCM that we must note carefully:

The Difference Between the Processes for Obtaining the GCF and the LCM
1. Notice the difference between step 2 for the LCM and step 2 for the GCF. For the GCF, we use only the bases that are common in the prime factorizations, whereas for the LCM, we use each base that appears in the prime factorizations.
2. Notice the difference between step 3 for the LCM and step 3 for the GCF. For the GCF, we attach the smallest exponents to the common bases, whereas for the LCM, we attach the largest exponents to the bases.

Sample Set C

Find the LCM of the following numbers.

9 and 12

1. $9=3⋅3=3^2$
$12=2⋅6 = 2⋅2⋅3 =2^2⋅3$
2. The bases that appear in the prime factorizations are 2 and 3.
3. The largest exponents appearing on 2 and 3 in the prime factorizations are, respectively, 2 and 2:
$2^2$ from 12.
$3^2$ from 9.
4. The LCM is the product of these numbers.
$\text{LCM }=2^2⋅3^2=4⋅9=36$

Thus, 36 is the smallest number that both 9 and 12 divide into without remainders.

90 and 630

1. $90 =2⋅45 =2⋅3 ⋅15 = 2⋅3⋅3⋅5 = 2⋅3^2⋅5$
$630 = 2⋅315=2⋅3⋅105=2⋅3⋅3⋅35=2⋅3⋅3⋅5⋅7$
$=2⋅3^2⋅5⋅7$
2. The bases that appear in the prime factorizations are 2, 3, 5, and 7.
3. The largest exponents that appear on 2, 3, 5, and 7 are, respectively, 1, 2, 1, and 1:
$2^1$ from either 90 or 630.
$3^2$ from either 90 or 630.
$5^1$ from either 90 or 630.
$7^1$ from 630.
4. The LCM is the product of these numbers.
$\text{LCM }=2⋅3^2⋅5⋅7=2⋅9⋅5⋅7=630$

Thus, 630 is the smallest number that both 90 and 630 divide into with no remainders.

33, 110, and 484

1. $33=3⋅11$
$110=2⋅55=2⋅5⋅11$
$484=2⋅242=2⋅2⋅121=2⋅2⋅11⋅11=2^2⋅11^2$.
2. The bases that appear in the prime factorizations are 2, 3, 5, and 11.
3. The largest exponents that appear on 2, 3, 5, and 11 are, respectively, 2, 1, 1, and 2:
$2^2$ from 484.
$3^1$ from 33.
$5^1$ from 110
$11^2$ from 484.
4. The LCM is the product of these numbers.
$\text{LCM =2^2⋅3⋅5⋅11^2}$
$= 4⋅3⋅5⋅121$
$=7260$

Thus, 7260 is the smallest number that 33, 110, and 484 divide into without remainders.

Source: Rice University, https://cnx.org/contents/XeVIW7Iw@4.6:0x3EJrKS@2/The-Least-Common-Multiple