Least Common Multiple Worksheets page - Free Printable

It looks like you've uploaded an answer key for a math worksheet titled "GCF & LCM Sheet 3 Answers", which lists the Greatest Common Factor (GCF) and Least Common Multiple (LCM) for pairs of numbers. However, since this is already an answer sheet, it appears that the original problem (the questions) was not included in your upload.

But I can help explain how to solve each problem on this sheet step by step, so you understand how the GCF and LCM were calculated. Let’s go through each pair:

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🔹 Definitions:

- GCF (Greatest Common Factor): The largest number that divides both numbers evenly.
- LCM (Least Common Multiple): The smallest number that is a multiple of both numbers.

We'll use prime factorization or listing factors/multiples to find these.

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✔ Problem-by-Problem Solution

#### 1) 40, 25
- Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
- Factors of 25: 1, 5, 25
- Common factors: 1, 5 → GCF = 5

- Multiples of 40: 40, 80, 120, 160, 200, ...
- Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, ...
- First common multiple: 200 → LCM = 200

✔ GCF = 5, LCM = 200

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#### 2) 24, 40
- Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
- Common: 1, 2, 4, 8 → GCF = 8

- Use formula:
$$
\text{LCM} = \frac{a \times b}{\text{GCF}} = \frac{24 \times 40}{8} = \frac{960}{8} = 120
$$
✔ GCF = 8, LCM = 120

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#### 3) 7, 15
- 7 is prime, 15 = 3×5 → only common factor is 1
- So GCF = 1

- LCM: Since they're coprime:
$$
\text{LCM} = 7 \times 15 = 105
$$
✔ GCF = 1, LCM = 105

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#### 4) 32, 16
- 32 = 2⁵, 16 = 2⁴ → common factor is 2⁴ = 16
- So GCF = 16

- LCM: Take highest powers: 2⁵ = 32
✔ GCF = 16, LCM = 32

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#### 5) 15, 9
- 15 = 3×5, 9 = 3² → common factor: 3 → GCF = 3

- LCM: 3² × 5 = 9 × 5 = 45
✔ GCF = 3, LCM = 45

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#### 6) 22, 33
- 22 = 2×11, 33 = 3×11 → common: 11 → GCF = 11

- LCM: 2×3×11 = 66
✔ GCF = 11, LCM = 66

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#### 7) 9, 20
- 9 = 3², 20 = 2²×5 → no common factors → GCF = 1

- LCM: 2² × 3² × 5 = 4 × 9 × 5 = 180
✔ GCF = 1, LCM = 180

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#### 8) 24, 30
- 24 = 2³×3, 30 = 2×3×5 → common: 2×3 = 6 → GCF = 6

- LCM: 2³ × 3 × 5 = 8 × 3 × 5 = 120
✔ GCF = 6, LCM = 120

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#### 9) 11, 15
- 11 is prime, 15 = 3×5 → no common factors → GCF = 1

- LCM = 11 × 15 = 165
✔ GCF = 1, LCM = 165

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#### 10) 50, 70
- 50 = 2×5², 70 = 2×5×7 → common: 2×5 = 10 → GCF = 10

- LCM: 2 × 5² × 7 = 2 × 25 × 7 = 350
✔ GCF = 10, LCM = 350

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✔ Summary Table:

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

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💡 Tips:

- For GCF: Use prime factorization and take the lowest power of common primes.
- For LCM: Take the highest power of all primes present.
- Or use:
$$
\text{LCM}(a,b) = \frac{a \times b}{\text{GCF}(a,b)}
$$

Let me know if you'd like a printable version, a worksheet with problems, or practice exercises!