2.10: Prime Factorization and the Least Common Multiple (Part 2)

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Find the Least Common Multiple (LCM) of Two Numbers

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.

Listing Multiples Method

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of 10 and 25. We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.

\[\begin{split} 10 & \colon \; 10, 20, 30, 40, \textbf{50}, 60, 70, 80, 90, \textbf{100}, 110, \ldots \\ 25 & \colon \; 25, \textbf{50}, 75, \textbf{100}, 125, \ldots \end{split} \nonumber \]

We see that $50$ and $100$ appear in both lists. They are common multiples of $10$ and $25$. We would find more common multiples if we continued the list of multiples for each.

The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of $10$ and $25$ is $50$.

HOW TO: FIND THE LEAST COMMON MULTIPLE (LCM) OF TWO NUMBERS BY LISTING MULTIPLES

Step 1. List the first several multiples of each number.

Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.

Step 3. Look for the smallest number that is common to both lists.

Step 4. This number is the LCM.

Example $\PageIndex{5}$: lcm

Find the LCM of $15$ and $20$ by listing multiples.

Solution

List the first several multiples of $15$ and of $20$. Identify the first common multiple.

\[\begin{split}15 & \colon \; 15, 30, 45, \textbf{60}, 75, 90, 105, 120 \\ 20 & \colon \; 20, 40, \textbf{60}, 80, 100, 120, 140, 160 \end{split} \nonumber\]

The smallest number to appear on both lists is $60$, so $60$ is the least common multiple of $15$ and $20$. Notice that $120$ is on both lists, too. It is a common multiple, but it is not the least common multiple.

Exercise $\PageIndex{9}$

Find the least common multiple (LCM) of the given numbers: $9$ and $12$

Answer: $36$

Exercise $\PageIndex{10}$

Find the least common multiple (LCM) of the given numbers: $18$ and $24$

Answer: $72$

Prime Factors Method

Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of $12$ and $18$.

We start by finding the prime factorization of each number.

\[12 = 2 \cdot 2 \cdot 3 \qquad \qquad 18 = 2 \cdot 3 \cdot 3 \nonumber\]

Then we write each number as a product of primes, matching primes vertically when possible.

\[\begin{split} 12 & = 2 \cdot 2 \cdot 3 \\ 18 & = 2 \cdot \quad \; 3 \cdot 3 \end{split} \nonumber \]

Now we bring down the primes in each column. The LCM is the product of these factors.

$The image shows the prime factorization of 12 written as the equation 12 equals 2 times 2 times 3. Below this equation is another showing the prime factorization of 18 written as the equation 18 equals 2 times 3 times 3. The two equations line up vertically at the equal symbol. The first 2 in the prime factorization of 12 aligns with the 2 in the prime factorization of 18. Under the second 2 in the prime factorization of 12 is a gap in the prime factorization of 18. Under the 3 in the prime factorization of 12 is the first 3 in the prime factorization of 18. The second 3 in the prime factorization has no factors above it from the prime factorization of 12. A horizontal line is drawn under the prime factorization of 18. Below this line is the equation LCM equal to 2 times 2 times 3 times 3. Arrows are drawn down vertically from the prime factorization of 12 through the prime factorization of 18 ending at the LCM equation. The first arrow starts at the first 2 in the prime factorization of 12 and continues down through the 2 in the prime factorization of 18. Ending with the first 2 in the LCM. The second arrow starts at the next 2 in the prime factorization of 12 and continues down through the gap in the prime factorization of 18. Ending with the second 2 in the LCM. The third arrow starts at the 3 in the prime factorization of 12 and continues down through the first 3 in the prime factorization of 18. Ending with the first 3 in the LCM. The last arrow starts at the second 3 in the prime factorization of 18 and points down to the second 3 in the LCM.$

Notice that the prime factors of $12$ and the prime factors of $18$ are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that $36$ is the least common multiple.

HOW TO: FIND THE LCM USING THE PRIME FACTORS METHOD

Step 1. Find the prime factorization of each number.

Step 2. Write each number as a product of primes, matching primes vertically when possible.

Step 3. Bring down the primes in each column.

Step 4. Multiply the factors to get the LCM.

Example $\PageIndex{6}$: lcm

Find the LCM of $15$ and $18$ using the prime factors method.

Solution

Write each number as a product of primes.	$15 = 3 \cdot 5 \qquad \qquad 18 = 2 \cdot 3 \cdot 3$
Write each number as a product of primes, matching primes vertically when possible.	$\begin{split} 15 & = \quad \; 3 \cdot \qquad 5 \\ 18 & = 2 \cdot 3 \cdot 3 \end{split}$
Bring down the primes in each column.	$CNX_BMath_Figure_02_05_026_img-03.png$
Multiply the factors to get the LCM.	LCM = 2 • 3 • 3 • 5 The LCM of 15 and 18 is 90.

Exercise $\PageIndex{11}$

Find the LCM using the prime factors method: $15$ and $20$

Answer: $60$

Exercise $\PageIndex{12}$

Find the LCM using the prime factors method: $15$ and $35$

Answer: $105$

Example $\PageIndex{7}$: lcm

Find the LCM of $50$ and $100$ using the prime factors method.

Solution

Write the prime factorization of each number.	$50 = 2 \cdot 5 \cdot 5 \qquad 100 = 2 \cdot 2 \cdot 5 \cdot 5$
Write each number as a product of primes, matching primes vertically when possible.	$\begin{split} 50 & = \quad \; 2 \cdot 5 \cdot 5 \\ 100 & = 2 \cdot 2 \cdot 5 \cdot 5 \end{split}$
Bring down the primes in each column.	$CNX_BMath_Figure_02_05_027_img-03.png$
Multiply the factors to get the LCM.	LCM = 2 • 2 • 5 • 5 The LCM of 50 and 100 is 100.

Exercise $\PageIndex{13}$

Find the LCM using the prime factors method: $55, 88$

Answer: $440$

Exercise $\PageIndex{14}$

Find the LCM using the prime factors method: $60, 72$

Answer: $360$

Access Additional Online Resources

Key Concepts

Find the prime factorization of a composite number using the tree method.
- Find any factor pair of the given number, and use these numbers to create two branches.
- If a factor is prime, that branch is complete. Circle the prime.
- If a factor is not prime, write it as the product of a factor pair and continue the process.
- Write the composite number as the product of all the circled primes.
Find the prime factorization of a composite number using the ladder method.
- Divide the number by the smallest prime.
- Continue dividing by that prime until it no longer divides evenly.
- Divide by the next prime until it no longer divides evenly.
- Continue until the quotient is a prime.
- Write the composite number as the product of all the primes on the sides and top of the ladder.
Find the LCM by listing multiples.
- List the first several multiples of each number.
- Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
- Look for the smallest number that is common to both lists.
- This number is the LCM.
Find the LCM using the prime factors method.
- Find the prime factorization of each number.
- Write each number as a product of primes, matching primes vertically when possible.
- Bring down the primes in each column.
- Multiply the factors to get the LCM.

Glossary

least common multiple: The smallest number that is a multiple of two numbers is called the least common multiple (LCM).
prime factorization: The prime factorization of a number is the product of prime numbers that equals the number.

Practice Makes Perfect

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using the factor tree method.

86
78
132
455
693
420
115
225
2475
1560

In the following exercises, find the prime factorization of each number using the ladder method.

56
72
168
252
391
400
432
627
2160
2520

In the following exercises, find the prime factorization of each number using any method.

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by listing multiples.

8, 12
4, 3
6, 15
12, 16
30, 40
20, 30
60, 75
44, 55

In the following exercises, find the least common multiple (LCM) by using the prime factors method.

8, 12
12, 16
24, 30
28, 40
70, 84
84, 90

In the following exercises, find the least common multiple (LCM) using any method.

6, 21
9, 15
24, 30
32, 40

Everyday Math

Grocery shopping Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)
Grocery shopping Paper plates are sold in packages of 12 and party cups come in packs of 8. What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)

Writing Exercises

Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Why?
Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why?

Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

$CNX_BMath_Figure_AppB_012.jpg$

(b) Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.

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HOW TO: FIND THE LEAST COMMON MULTIPLE (LCM) OF TWO NUMBERS BY LISTING MULTIPLES

Example \(\PageIndex{5}\): lcm

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)

HOW TO: FIND THE LCM USING THE PRIME FACTORS METHOD

Example \(\PageIndex{6}\): lcm

Exercise \(\PageIndex{11}\)

Exercise \(\PageIndex{12}\)

Example \(\PageIndex{7}\): lcm

Exercise \(\PageIndex{13}\)

Exercise \(\PageIndex{14}\)

Write each number as a product of primes.	\(15 = 3 \cdot 5 \qquad \qquad 18 = 2 \cdot 3 \cdot 3\)
Write each number as a product of primes, matching primes vertically when possible.	\(\begin{split} 15 & = \quad \; 3 \cdot \qquad 5 \\ 18 & = 2 \cdot 3 \cdot 3 \end{split}\)
Bring down the primes in each column.	$CNX_BMath_Figure_02_05_026_img-03.png$
Multiply the factors to get the LCM.	LCM = 2 • 3 • 3 • 5 The LCM of 15 and 18 is 90.

Write the prime factorization of each number.	\(50 = 2 \cdot 5 \cdot 5 \qquad 100 = 2 \cdot 2 \cdot 5 \cdot 5\)
Write each number as a product of primes, matching primes vertically when possible.	\(\begin{split} 50 & = \quad \; 2 \cdot 5 \cdot 5 \\ 100 & = 2 \cdot 2 \cdot 5 \cdot 5 \end{split}\)
Bring down the primes in each column.	$CNX_BMath_Figure_02_05_027_img-03.png$
Multiply the factors to get the LCM.	LCM = 2 • 2 • 5 • 5 The LCM of 50 and 100 is 100.