GCF & LCM Sheet 3: A math worksheet for calculating greatest common factors and least common multiples.

To solve the problems on the "GCF & LCM SHEET 3," we need to find both the Greatest Common Factor (GCF) and the Least Common Multiple (LCM) for each pair of numbers. Let's go through each problem step by step.

1. \( 40, 25 \)

#### GCF:
- Prime factorization:
- \( 40 = 2^3 \times 5 \)
- \( 25 = 5^2 \)
- Common factors: \( 5 \)
- GCF: \( 5 \)

#### LCM:
- LCM is the product of the highest powers of all prime factors:
- \( 2^3 \times 5^2 = 8 \times 25 = 200 \)
- LCM: \( 200 \)

Answer:
- GCF: \( 5 \)
- LCM: \( 200 \)

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2. \( 24, 40 \)

#### GCF:
- Prime factorization:
- \( 24 = 2^3 \times 3 \)
- \( 40 = 2^3 \times 5 \)
- Common factors: \( 2^3 = 8 \)
- GCF: \( 8 \)

#### LCM:
- LCM is the product of the highest powers of all prime factors:
- \( 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120 \)
- LCM: \( 120 \)

Answer:
- GCF: \( 8 \)
- LCM: \( 120 \)

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3. \( 7, 15 \)

#### GCF:
- Prime factorization:
- \( 7 = 7 \)
- \( 15 = 3 \times 5 \)
- No common factors other than \( 1 \)
- GCF: \( 1 \)

#### LCM:
- LCM is the product of all prime factors since they are coprime:
- \( 7 \times 15 = 105 \)
- LCM: \( 105 \)

Answer:
- GCF: \( 1 \)
- LCM: \( 105 \)

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4. \( 32, 16 \)

#### GCF:
- Prime factorization:
- \( 32 = 2^5 \)
- \( 16 = 2^4 \)
- Common factors: \( 2^4 = 16 \)
- GCF: \( 16 \)

#### LCM:
- LCM is the product of the highest powers of all prime factors:
- \( 2^5 = 32 \)
- LCM: \( 32 \)

Answer:
- GCF: \( 16 \)
- LCM: \( 32 \)

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5. \( 15, 9 \)

#### GCF:
- Prime factorization:
- \( 15 = 3 \times 5 \)
- \( 9 = 3^2 \)
- Common factors: \( 3 \)
- GCF: \( 3 \)

#### LCM:
- LCM is the product of the highest powers of all prime factors:
- \( 3^2 \times 5 = 9 \times 5 = 45 \)
- LCM: \( 45 \)

Answer:
- GCF: \( 3 \)
- LCM: \( 45 \)

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6. \( 22, 33 \)

#### GCF:
- Prime factorization:
- \( 22 = 2 \times 11 \)
- \( 33 = 3 \times 11 \)
- Common factors: \( 11 \)
- GCF: \( 11 \)

#### LCM:
- LCM is the product of the highest powers of all prime factors:
- \( 2 \times 3 \times 11 = 66 \)
- LCM: \( 66 \)

Answer:
- GCF: \( 11 \)
- LCM: \( 66 \)

---

7. \( 9, 20 \)

#### GCF:
- Prime factorization:
- \( 9 = 3^2 \)
- \( 20 = 2^2 \times 5 \)
- No common factors other than \( 1 \)
- GCF: \( 1 \)

#### LCM:
- LCM is the product of all prime factors:
- \( 3^2 \times 2^2 \times 5 = 9 \times 4 \times 5 = 180 \)
- LCM: \( 180 \)

Answer:
- GCF: \( 1 \)
- LCM: \( 180 \)

---

8. \( 24, 30 \)

#### GCF:
- Prime factorization:
- \( 24 = 2^3 \times 3 \)
- \( 30 = 2 \times 3 \times 5 \)
- Common factors: \( 2 \times 3 = 6 \)
- GCF: \( 6 \)

#### LCM:
- LCM is the product of the highest powers of all prime factors:
- \( 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120 \)
- LCM: \( 120 \)

Answer:
- GCF: \( 6 \)
- LCM: \( 120 \)

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9. \( 11, 15 \)

#### GCF:
- Prime factorization:
- \( 11 = 11 \)
- \( 15 = 3 \times 5 \)
- No common factors other than \( 1 \)
- GCF: \( 1 \)

#### LCM:
- LCM is the product of all prime factors:
- \( 11 \times 15 = 165 \)
- LCM: \( 165 \)

Answer:
- GCF: \( 1 \)
- LCM: \( 165 \)

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10. \( 50, 70 \)

#### GCF:
- Prime factorization:
- \( 50 = 2 \times 5^2 \)
- \( 70 = 2 \times 5 \times 7 \)
- Common factors: \( 2 \times 5 = 10 \)
- GCF: \( 10 \)

#### LCM:
- LCM is the product of the highest powers of all prime factors:
- \( 2 \times 5^2 \times 7 = 2 \times 25 \times 7 = 350 \)
- LCM: \( 350 \)

Answer:
- GCF: \( 10 \)
- LCM: \( 350 \)

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Final Answers:

| Pair | GCF | LCM |
|------------|-------|-------|
| 1. \( 40, 25 \) | 5 | 200 |
| 2. \( 24, 40 \) | 8 | 120 |
| 3. \( 7, 15 \) | 1 | 105 |
| 4. \( 32, 16 \) | 16 | 32 |
| 5. \( 15, 9 \) | 3 | 45 |
| 6. \( 22, 33 \) | 11 | 66 |
| 7. \( 9, 20 \) | 1 | 180 |
| 8. \( 24, 30 \) | 6 | 120 |
| 9. \( 11, 15 \) | 1 | 165 |
| 10. \( 50, 70 \)| 10 | 350 |

\boxed{
\begin{array}{|c|c|c|}
\hline
\text{Pair} & \text{GCF} & \text{LCM} \\
\hline
1. \ 40, 25 & 5 & 200 \\
2. \ 24, 40 & 8 & 120 \\
3. \ 7, 15 & 1 & 105 \\
4. \ 32, 16 & 16 & 32 \\
5. \ 15, 9 & 3 & 45 \\
6. \ 22, 33 & 11 & 66 \\
7. \ 9, 20 & 1 & 180 \\
8. \ 24, 30 & 6 & 120 \\
9. \ 11, 15 & 1 & 165 \\
10. \ 50, 70 & 10 & 350 \\
\hline
\end{array}
}