Least Common Multiples Worksheets - 15 Worksheets Library - Free Printable

To solve the problem of finding the Least Common Multiple (LCM) for each set of numbers, we will use the prime factorization method. Here's a step-by-step explanation and solution for each pair:

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Step 1: Understand the LCM

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers. To find the LCM using prime factorization:
1. Find the prime factors of each number.
2. For each distinct prime factor, take the highest power that appears in any of the factorizations.
3. Multiply these highest powers together to get the LCM.

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Solutions

#### Problem 1: LCM (20, 10, 30)
- Prime factorization:
- \( 20 = 2^2 \times 5 \)
- \( 10 = 2 \times 5 \)
- \( 30 = 2 \times 3 \times 5 \)
- Highest powers of each prime:
- \( 2^2 \) (from 20)
- \( 3^1 \) (from 30)
- \( 5^1 \) (common in all)
- LCM = \( 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \)

Answer: 60

#### Problem 2: LCM (15, 25, 30)
- Prime factorization:
- \( 15 = 3 \times 5 \)
- \( 25 = 5^2 \)
- \( 30 = 2 \times 3 \times 5 \)
- Highest powers of each prime:
- \( 2^1 \) (from 30)
- \( 3^1 \) (from 15 and 30)
- \( 5^2 \) (from 25)
- LCM = \( 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \)

Answer: 150

#### Problem 3: LCM (12, 36, 20)
- Prime factorization:
- \( 12 = 2^2 \times 3 \)
- \( 36 = 2^2 \times 3^2 \)
- \( 20 = 2^2 \times 5 \)
- Highest powers of each prime:
- \( 2^2 \) (common in all)
- \( 3^2 \) (from 36)
- \( 5^1 \) (from 20)
- LCM = \( 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \)

Answer: 180

#### Problem 4: LCM (25, 40, 30)
- Prime factorization:
- \( 25 = 5^2 \)
- \( 40 = 2^3 \times 5 \)
- \( 30 = 2 \times 3 \times 5 \)
- Highest powers of each prime:
- \( 2^3 \) (from 40)
- \( 3^1 \) (from 30)
- \( 5^2 \) (from 25)
- LCM = \( 2^3 \times 3^1 \times 5^2 = 8 \times 3 \times 25 = 600 \)

Answer: 600

#### Problem 5: LCM (25, 20, 15)
- Prime factorization:
- \( 25 = 5^2 \)
- \( 20 = 2^2 \times 5 \)
- \( 15 = 3 \times 5 \)
- Highest powers of each prime:
- \( 2^2 \) (from 20)
- \( 3^1 \) (from 15)
- \( 5^2 \) (from 25)
- LCM = \( 2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 = 300 \)

Answer: 300

#### Problem 6: LCM (36, 12, 45)
- Prime factorization:
- \( 36 = 2^2 \times 3^2 \)
- \( 12 = 2^2 \times 3 \)
- \( 45 = 3^2 \times 5 \)
- Highest powers of each prime:
- \( 2^2 \) (from 36 and 12)
- \( 3^2 \) (from 36 and 45)
- \( 5^1 \) (from 45)
- LCM = \( 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180 \)

Answer: 180

#### Problem 7: LCM (24, 9, 15)
- Prime factorization:
- \( 24 = 2^3 \times 3 \)
- \( 9 = 3^2 \)
- \( 15 = 3 \times 5 \)
- Highest powers of each prime:
- \( 2^3 \) (from 24)
- \( 3^2 \) (from 9)
- \( 5^1 \) (from 15)
- LCM = \( 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360 \)

Answer: 360

#### Problem 8: LCM (20, 10, 15)
- Prime factorization:
- \( 20 = 2^2 \times 5 \)
- \( 10 = 2 \times 5 \)
- \( 15 = 3 \times 5 \)
- Highest powers of each prime:
- \( 2^2 \) (from 20)
- \( 3^1 \) (from 15)
- \( 5^1 \) (common in all)
- LCM = \( 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60 \)

Answer: 60

#### Problem 9: LCM (24, 36, 12)
- Prime factorization:
- \( 24 = 2^3 \times 3 \)
- \( 36 = 2^2 \times 3^2 \)
- \( 12 = 2^2 \times 3 \)
- Highest powers of each prime:
- \( 2^3 \) (from 24)
- \( 3^2 \) (from 36)
- LCM = \( 2^3 \times 3^2 = 8 \times 9 = 72 \)

Answer: 72

#### Problem 10: LCM (45, 25, 50)
- Prime factorization:
- \( 45 = 3^2 \times 5 \)
- \( 25 = 5^2 \)
- \( 50 = 2 \times 5^2 \)
- Highest powers of each prime:
- \( 2^1 \) (from 50)
- \( 3^2 \) (from 45)
- \( 5^2 \) (from 25 and 50)
- LCM = \( 2^1 \times 3^2 \times 5^2 = 2 \times 9 \times 25 = 450 \)

Answer: 450

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Final Answers

1. \( \boxed{60} \)
2. \( \boxed{150} \)
3. \( \boxed{180} \)
4. \( \boxed{600} \)
5. \( \boxed{300} \)
6. \( \boxed{180} \)
7. \( \boxed{360} \)
8. \( \boxed{60} \)
9. \( \boxed{72} \)
10. \( \boxed{450} \)

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These are the final answers for each problem. Each step involves identifying the prime factors and then combining them to find the LCM.