Lesson 57: Finding the Least Common Multiple

New Concept

The least number that is evenly divisible by each of the numbers in a set of numbers is called the least common multiple or LCM.

What’s next

Next, you'll use prime factorization to find the LCM for sets of numbers and then for algebraic expressions.

Property

The least number that is evenly divisible by each of the numbers in a set of numbers. To find it, write each number as a product of prime factors. Then, use every prime factor the greatest number of times it appears in any single number's factorization.

Explanation

Think of it like building the ultimate playlist from your friends' favorite songs, which are the prime factors. Your LCM playlist must include the maximum number of times any one friend listed a particular song. This way, everyone's top hit is covered, and the playlist is as short as possible while still pleasing the whole group.

Examples

Find the LCM of 24 and 36. We have $24 = 2^3 \cdot 3$ and $36 = 2^2 \cdot 3^2$. The LCM needs the highest powers: $2^3 \cdot 3^2 = 72$.
Find the LCM of 11, 12, and 18. We have $11=11$, $12 = 2^2 \cdot 3$, and $18 = 2 \cdot 3^2$. The LCM is $2^2 \cdot 3^2 \cdot 11 = 396$.

Finding the smallest number two values share can be simple if you break them down first. Let's practice the first key idea from this lesson, identifying the LCM of a Set of Numbers.

Example Problem
Find the LCM of 30 and 45.

Step-by-Step

1. First, write each number as a product of its prime factors.

2. The number 2 is a factor of 30. It appears once, so it will appear once in the LCM.

3. The number 3 is a factor of both numbers. The greatest number of times it appears is two times in 45, so it will appear in the LCM two times.

4. The number 5 is a factor of both numbers, appearing one time in each. So, it will appear once in the LCM.

5. Now, multiply the factors to find the LCM.

The LCM of 30 and 45 is 90.

Property

To find the LCM of monomials, find the LCM of the coefficients. Then, for each variable, take the highest power that appears in any of the terms. The LCM is the product of the numerical LCM and the variables raised to their highest powers.

Explanation

This is just like our playlist example, but now we're adding different genres, which are the variables! Treat each variable like a unique artist. For your final LCM, you need the LCM of the numbers, or coefficients, and the "greatest hits" version of each artist, which means taking the highest power of each variable you can find.

Examples

Find the LCM of $10a^3c^5$ and $15a^2c^4$. The LCM of 10 and 15 is 30. The highest power of $a$ is $a^3$ and of $c$ is $c^5$. So, the LCM is $30a^3c^5$.
Find the LCM of $6p^2s^3$ and $8m^3p$. The LCM of 6 and 8 is 24. The highest powers are $p^2$, $s^3$, and $m^3$. The LCM is $24m^3p^2s^3$.

Factoring polynomials is the secret to finding their LCM. Let's try this with the second key idea from this lesson, identifying the LCM of polynomials.

Example Problem
Find the LCM of $(8x^2+24x)$ and $(10x+30)$.

Step-by-Step

1. First, factor each binomial to find its prime components.
2. The Greatest Common Factor (GCF) of the terms in $(8x^2+24x)$ is $8x$. Factor it out.

3. The GCF of the terms in $(10x+30)$ is $10$. Factor it out.

4. Now, identify all the unique factors. We have the numeric factors 2 and 5, the variable factor $x$, and the binomial factor $(x+3)$.
5. To build the LCM, take the greatest number of times each factor appears in any single expression. The factor 2 appears most (three times). The factor 5 appears once. The factor $x$ appears once. The factor $(x+3)$ appears once in each.

6. Multiply these factors to get the final LCM.

The LCM is $40x(x+3)$.