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.
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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 \(50\).
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.
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. See more: "Uh Huh, You Got The Right One Baby Uh Huh, You Got The Right One, Baby, Uh | LCM = 2 • 2 • 5 • 5 The LCM of 50 and 100 is 100. |
Exercise \(\PageIndex{14}\)
Find the LCM using the prime factors method: \(60, 72\)
Answer\(360\)