Math worksheet for finding LCM and GCF using prime factorization.

To solve the problems of finding the Least Common Multiple (LCM) and Greatest Common Factor (GCF) using prime factorization, we will follow these steps:

Step-by-Step Process:

1. Prime Factorization: Break down each number into its prime factors.
2. Greatest Common Factor (GCF):
- Identify the common prime factors among all the numbers.
- Take the lowest power of each common prime factor.
3. Least Common Multiple (LCM):
- List all the prime factors that appear in any of the numbers.
- For each prime factor, take the highest power that appears in any of the numbers.

Solutions for Each Problem:

#### Problem 1: 4, 5, 24
- Prime Factorization:
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 24 = 2^3 \times 3^1 \)

- GCF:
- No common prime factors other than 1.
- \( \text{GCF} = 1 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^3 \) (from 24)
- \( 3^1 \) (from 24)
- \( 5^1 \) (from 5)
- \( \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \)

- Answer:
- \( \text{GCF} = 1 \)
- \( \text{LCM} = 120 \)

#### Problem 2: 4, 15, 16
- Prime Factorization:
- \( 4 = 2^2 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 16 = 2^4 \)

- GCF:
- No common prime factors other than 1.
- \( \text{GCF} = 1 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^4 \) (from 16)
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 15)
- \( \text{LCM} = 2^4 \times 3^1 \times 5^1 = 16 \times 3 \times 5 = 240 \)

- Answer:
- \( \text{GCF} = 1 \)
- \( \text{LCM} = 240 \)

#### Problem 3: 32, 34, 36
- Prime Factorization:
- \( 32 = 2^5 \)
- \( 34 = 2^1 \times 17^1 \)
- \( 36 = 2^2 \times 3^2 \)

- GCF:
- The only common prime factor is \( 2 \) with the lowest power being \( 2^1 \).
- \( \text{GCF} = 2 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^5 \) (from 32)
- \( 3^2 \) (from 36)
- \( 17^1 \) (from 34)
- \( \text{LCM} = 2^5 \times 3^2 \times 17^1 = 32 \times 9 \times 17 = 4896 \)

- Answer:
- \( \text{GCF} = 2 \)
- \( \text{LCM} = 4896 \)

#### Problem 4: 34, 35, 32
- Prime Factorization:
- \( 34 = 2^1 \times 17^1 \)
- \( 35 = 5^1 \times 7^1 \)
- \( 32 = 2^5 \)

- GCF:
- No common prime factors other than 1.
- \( \text{GCF} = 1 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^5 \) (from 32)
- \( 5^1 \) (from 35)
- \( 7^1 \) (from 35)
- \( 17^1 \) (from 34)
- \( \text{LCM} = 2^5 \times 5^1 \times 7^1 \times 17^1 = 32 \times 5 \times 7 \times 17 = 19040 \)

- Answer:
- \( \text{GCF} = 1 \)
- \( \text{LCM} = 19040 \)

#### Problem 5: 21, 24, 36
- Prime Factorization:
- \( 21 = 3^1 \times 7^1 \)
- \( 24 = 2^3 \times 3^1 \)
- \( 36 = 2^2 \times 3^2 \)

- GCF:
- The common prime factor is \( 3 \) with the lowest power being \( 3^1 \).
- \( \text{GCF} = 3 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^3 \) (from 24)
- \( 3^2 \) (from 36)
- \( 7^1 \) (from 21)
- \( \text{LCM} = 2^3 \times 3^2 \times 7^1 = 8 \times 9 \times 7 = 504 \)

- Answer:
- \( \text{GCF} = 3 \)
- \( \text{LCM} = 504 \)

#### Problem 6: 10, 16, 21
- Prime Factorization:
- \( 10 = 2^1 \times 5^1 \)
- \( 16 = 2^4 \)
- \( 21 = 3^1 \times 7^1 \)

- GCF:
- No common prime factors other than 1.
- \( \text{GCF} = 1 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^4 \) (from 16)
- \( 3^1 \) (from 21)
- \( 5^1 \) (from 10)
- \( 7^1 \) (from 21)
- \( \text{LCM} = 2^4 \times 3^1 \times 5^1 \times 7^1 = 16 \times 3 \times 5 \times 7 = 1680 \)

- Answer:
- \( \text{GCF} = 1 \)
- \( \text{LCM} = 1680 \)

#### Problem 7: 21, 36, 32
- Prime Factorization:
- \( 21 = 3^1 \times 7^1 \)
- \( 36 = 2^2 \times 3^2 \)
- \( 32 = 2^5 \)

- GCF:
- No common prime factors other than 1.
- \( \text{GCF} = 1 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^5 \) (from 32)
- \( 3^2 \) (from 36)
- \( 7^1 \) (from 21)
- \( \text{LCM} = 2^5 \times 3^2 \times 7^1 = 32 \times 9 \times 7 = 2016 \)

- Answer:
- \( \text{GCF} = 1 \)
- \( \text{LCM} = 2016 \)

#### Problem 8: 34, 21, 14
- Prime Factorization:
- \( 34 = 2^1 \times 17^1 \)
- \( 21 = 3^1 \times 7^1 \)
- \( 14 = 2^1 \times 7^1 \)

- GCF:
- The only common prime factor is \( 7 \) with the lowest power being \( 7^1 \).
- \( \text{GCF} = 7 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^1 \) (from 34 and 14)
- \( 3^1 \) (from 21)
- \( 7^1 \) (from 21 and 14)
- \( 17^1 \) (from 34)
- \( \text{LCM} = 2^1 \times 3^1 \times 7^1 \times 17^1 = 2 \times 3 \times 7 \times 17 = 714 \)

- Answer:
- \( \text{GCF} = 7 \)
- \( \text{LCM} = 714 \)

#### Problem 9: 35, 15, 12
- Prime Factorization:
- \( 35 = 5^1 \times 7^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 12 = 2^2 \times 3^1 \)

- GCF:
- The only common prime factor is \( 5 \) with the lowest power being \( 5^1 \).
- \( \text{GCF} = 5 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^2 \) (from 12)
- \( 3^1 \) (from 15 and 12)
- \( 5^1 \) (from 35 and 15)
- \( 7^1 \) (from 35)
- \( \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420 \)

- Answer:
- \( \text{GCF} = 5 \)
- \( \text{LCM} = 420 \)

#### Problem 10: 34, 15, 4
- Prime Factorization:
- \( 34 = 2^1 \times 17^1 \)
- \( 15 = 3^1 \times 5^1 \)
- \( 4 = 2^2 \)

- GCF:
- No common prime factors other than 1.
- \( \text{GCF} = 1 \)

- LCM:
- Combine all prime factors with their highest powers:
- \( 2^2 \) (from 4)
- \( 3^1 \) (from 15)
- \( 5^1 \) (from 15)
- \( 17^1 \) (from 34)
- \( \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 17^1 = 4 \times 3 \times 5 \times 17 = 1020 \)

- Answer:
- \( \text{GCF} = 1 \)
- \( \text{LCM} = 1020 \)

Final Answers:

1. \( \boxed{120, 1} \)
2. \( \boxed{240, 1} \)
3. \( \boxed{4896, 2} \)
4. \( \boxed{19040, 1} \)
5. \( \boxed{504, 3} \)
6. \( \boxed{1680, 1} \)
7. \( \boxed{2016, 1} \)
8. \( \boxed{714, 7} \)
9. \( \boxed{420, 5} \)
10. \( \boxed{1020, 1} \)