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.
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. |
|
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. |
|
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
-
\(360\)
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.
- 150
- 180
- 525
- 444
- 36
- 50
- 350
- 144
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
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?