Instructions
The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. Think of it as the first number you'd hit if you were skip-counting by each of your starting numbers at the same time. Let's explore two ways to find it!
Method 1: The Listing Method
This is the most straightforward method. You simply list the multiples of each number until you find the first one that appears in all lists. This method is great for smaller numbers.
Example: Find the LCM of 4 and 6.
- Multiples of 4: 4, 8, 12, 16, 20, 24...
- Multiples of 6: 6, 12, 18, 24, 30...
The first number they have in common is 12. So, the LCM of 4 and 6 is 12.
Your Turn! Use the listing method to find the LCM for the following pairs.
1. Find the LCM of 5 and 10.
2. Find the LCM of 8 and 12.
3. Find the LCM of 3 and 7.
Method 2: The Prime Factorization Method
This method is more powerful, especially for larger numbers. It might seem like more steps, but it's often faster in the long run.
- Find the prime factorization of each number. A factor tree can help!
- Write down all the prime factors that appear in any of the factorizations.
- For each prime factor, find the highest power (the most times it appears) in any one factorization.
- Multiply these highest powers together to get the LCM.
Example: Find the LCM of 18 and 24.
- Prime factorization of 18: 2 × 3 × 3 =
2¹ × 3² - Prime factorization of 24: 2 × 2 × 2 × 3 =
2³ × 3¹
The prime factors involved are 2 and 3.
- The highest power of 2 is
2³(from 24). - The highest power of 3 is
3²(from 18).
Now, multiply them together: LCM = 2³ × 3² = 8 × 9 = 72.
Your Turn! Use the prime factorization method to find the LCM.
4. Find the LCM of 20 and 50.
5. Find the LCM of 28 and 42.
6. Find the LCM of 15 and 35.
Challenge Problems
Now, solve these problems using whichever method you prefer. Show your work!
7. Find the LCM of 9 and 15.
8. Find the LCM of three numbers: 4, 5, and 6.
9. Word Problem: Hot dogs come in packs of 10, and hot dog buns come in packs of 8. What is the smallest number of full packs of each you need to buy to have exactly the same number of hot dogs and buns?
Answer Key
- 10 (Multiples of 5: 5, 10...; Multiples of 10: 10...)
- 24 (Multiples of 8: 8, 16, 24...; Multiples of 12: 12, 24...)
- 21 (Multiples of 3: 3, 6, 9, 12, 15, 18, 21...; Multiples of 7: 7, 14, 21...)
- 100 (Prime factors of 20 = 2² × 5; Prime factors of 50 = 2 × 5². LCM = 2² × 5² = 4 × 25 = 100)
- 84 (Prime factors of 28 = 2² × 7; Prime factors of 42 = 2 × 3 × 7. LCM = 2² × 3 × 7 = 4 × 3 × 7 = 84)
- 105 (Prime factors of 15 = 3 × 5; Prime factors of 35 = 5 × 7. LCM = 3 × 5 × 7 = 105)
- 45
- 60
- The LCM of 10 and 8 is 40. You would need to buy 4 packs of hot dogs (4 × 10 = 40) and 5 packs of buns (5 × 8 = 40).