Least Common Multiple: Methods and Examples

What Is Least Common Multiple?

When we list the multiples of two numbers side by side, they will share many values. These shared values are their common multiples.

The least common multiple (LCM) is the very first, or smallest, number that appears in both lists. Also known as the lowest common multiple, this concept guarantees you have found the smallest possible quantity that can be evenly divided by your starting numbers.

The LCM is never smaller than the numbers you are comparing. If one number happens to be a multiple of the other, the larger number itself is the LCM.

When to Use It

The LCM is frequently used to find common denominators when adding or subtracting fractions. It is also the perfect mathematical tool to solve real-world problems involving repeating events, such as finding out when two different blinking lights or bus schedules will sync up at the exact same time.

If two numbers share no prime factors in common other than $1$, they are coprime numbers. In this case, their LCM is simply their product. After mastering this concept, you can explore the relationship between GCF and LCM to solve more advanced divisibility problems.

Step-by-Step Method

There are three main methods for finding the LCM. Choose the one that works best for the size of the numbers you are given.

Listing Method

1. List the first several multiples of each number.
2. Identify the smallest multiple that appears in every list.

Prime Factorization Method

1. Write each number as a product of prime numbers.
2. Identify the highest power of each prime factor present.
3. Multiply these highest powers together to find the LCM.

Division (Ladder) Method

1. Write the numbers in a row.
2. Divide by a common prime factor or their greatest common factor.
3. Write the results in the row below. If a number is not divisible, bring it down unchanged.
4. Repeat until the numbers in the bottom row have no common factors other than $1$.
5. Multiply all the divisors on the left and the numbers in the bottom row to find the LCM.

Visual Worked Examples

Review these three scenarios to see how each method helps you find the correct answer efficiently.

Example 1: Listing multiples

Question: What is the least common multiple of $6$ and $8$?

Method:

1. List the multiples of $6$: $6, 12, 18, 24, 30 \dots$
2. List the multiples of $8$: $8, 16, 24, 32 \dots$
3. Find the smallest number present in both lists.

Answer: The least common multiple is $24$.

Check: $24 \div 6 = 4$ and $24 \div 8 = 3$. Because both divide evenly with no remainder, $24$ is a valid common multiple. Since no smaller number in the lists matched, it is the least common multiple.

Example 2: Prime factorization method

Question: Find the LCM of $12$ and $18$ using prime factorization.

Method:

1. Break $12$ into its prime factors: $12 = 2 \times 2 \times 3 = 2^2 \times 3^1$.
2. Break $18$ into its prime factors: $18 = 2 \times 3 \times 3 = 2^1 \times 3^2$.
3. Take the highest power of each prime factor. For $2$, the highest power is $2^2$. For $3$, the highest power is $3^2$.
4. Multiply these highest powers together: $2^2 \times 3^2 = 4 \times 9$.

Answer: The LCM is $36$.

Check: $36 \div 12 = 3$ and $36 \div 18 = 2$.

Example 3: Division method and repeating events

Question: Bus A arrives every $24$ minutes. Bus B arrives every $30$ minutes. If they arrive together now, in how many minutes will they next arrive at the exact same time?

Method:

1. Finding the next simultaneous arrival requires the LCM of $24$ and $30$.
2. Write $24$ and $30$ in a row. Divide both by a common factor, such as $2$. This leaves $12$ and $15$.
3. Divide $12$ and $15$ by a common factor of $3$. This leaves $4$ and $5$.
4. The remaining numbers, $4$ and $5$, share no common factors other than $1$.
5. Multiply the outer numbers forming the highlighted shape: the divisors $2$ and $3$, and the bottom numbers $4$ and $5$.
6. $2 \times 3 \times 4 \times 5 = 120$.

Answer: They will arrive together again in $120$ minutes.

Check: $120 \div 24 = 5$ trips for Bus A, and $120 \div 30 = 4$ trips for Bus B.

How to Check the Answer

To verify that your LCM is mathematically correct, divide it by each of the starting numbers. If any division results in a remainder or a decimal, the number is not a valid common multiple.

To guarantee it is the least common multiple, ensure that the final quotients you receive share no common prime factors.

Common Mistakes

Confusing the concepts

The greatest common factor is the largest number that divides into your starting numbers. The least common multiple is the smallest number that your starting numbers divide into. The LCM is always equal to or larger than the original numbers, while the factor is always equal to or smaller.

Always multiplying the numbers together

Multiplying the two numbers directly will always give you a common multiple, but it is often not the least common multiple. For example, multiplying $4 \times 6$ gives $24$, but their actual LCM is $12$. Only multiply the numbers directly when they have no shared prime factors.