GCF and LCM - Free Printable

To solve the problem of finding the Greatest Common Factor (GCF) and Least Common Multiple (LCM) for each pair of numbers, we will use the following methods:

Steps to Find GCF:

1. Prime Factorization: Break down each number into its prime factors.
2. Identify Common Factors: Find the common prime factors of both numbers.
3. Multiply Common Factors: Multiply the lowest powers of all common prime factors to get the GCF.

Steps to Find LCM:

1. Prime Factorization: Break down each number into its prime factors.
2. Identify All Prime Factors: Include all prime factors from both numbers.
3. Take the Highest Powers: For each prime factor, take the highest power that appears in either factorization.
4. Multiply: Multiply these highest powers together to get the LCM.

Now, let's solve each pair step by step.

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1) 40, 25

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

#### LCM:
- Prime factors: \( 2^3, 5^2 \)
- LCM = \( 2^3 \times 5^2 = 8 \times 25 = 200 \)

Result:
- GCF = 5
- LCM = 200

---

2) 24, 40

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

#### LCM:
- Prime factors: \( 2^3, 3, 5 \)
- LCM = \( 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120 \)

Result:
- GCF = 8
- LCM = 120

---

3) 7, 15

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

#### LCM:
- Prime factors: \( 7, 3, 5 \)
- LCM = \( 7 \times 3 \times 5 = 105 \)

Result:
- GCF = 1
- LCM = 105

---

4) 32, 16

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

#### LCM:
- Prime factors: \( 2^5 \)
- LCM = \( 2^5 = 32 \)

Result:
- GCF = 16
- LCM = 32

---

5) 15, 9

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

#### LCM:
- Prime factors: \( 3^2, 5 \)
- LCM = \( 3^2 \times 5 = 9 \times 5 = 45 \)

Result:
- GCF = 3
- LCM = 45

---

6) 22, 33

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

#### LCM:
- Prime factors: \( 2, 3, 11 \)
- LCM = \( 2 \times 3 \times 11 = 66 \)

Result:
- GCF = 11
- LCM = 66

---

7) 9, 20

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

#### LCM:
- Prime factors: \( 3^2, 2^2, 5 \)
- LCM = \( 3^2 \times 2^2 \times 5 = 9 \times 4 \times 5 = 180 \)

Result:
- GCF = 1
- LCM = 180

---

8) 24, 30

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

#### LCM:
- Prime factors: \( 2^3, 3, 5 \)
- LCM = \( 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120 \)

Result:
- GCF = 6
- LCM = 120

---

9) 11, 15

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

#### LCM:
- Prime factors: \( 11, 3, 5 \)
- LCM = \( 11 \times 3 \times 5 = 165 \)

Result:
- GCF = 1
- LCM = 165

---

10) 50, 70

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

#### LCM:
- Prime factors: \( 2, 5^2, 7 \)
- LCM = \( 2 \times 5^2 \times 7 = 2 \times 25 \times 7 = 350 \)

Result:
- GCF = 10
- LCM = 350

---

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