Least Common Multiples Worksheets - 15 Worksheets Library - Free Printable

Let's solve each of the Least Common Multiple (LCM) problems step by step using prime factorization, which is the method suggested by the factor tree diagrams in the worksheet.

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Problem 1: LCM of 50 and 75

#### Step 1: Prime Factorization
- 50:
$$
50 = 2 \times 25 = 2 \times 5 \times 5 = 2^1 \times 5^2
$$

- 75:
$$
75 = 3 \times 25 = 3 \times 5 \times 5 = 3^1 \times 5^2
$$

#### Step 2: Take the highest power of each prime factor:
- $2^1$ (from 50)
- $3^1$ (from 75)
- $5^2$ (common in both)

#### Step 3: Multiply them together:
$$
LCM = 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150
$$

✔ Answer: LCM = 150

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Problem 2: LCM of 40 and 60

#### Step 1: Prime Factorization
- 40:
$$
40 = 2 \times 20 = 2 \times 2 \times 10 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1
$$

- 60:
$$
60 = 2 \times 30 = 2 \times 2 \times 15 = 2 \times 2 \times 3 \times 5 = 2^2 \times 3^1 \times 5^1
$$

#### Step 2: Highest powers:
- $2^3$ (from 40)
- $3^1$ (from 60)
- $5^1$ (common)

#### Step 3: Multiply:
$$
LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120
$$

✔ Answer: LCM = 120

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Problem 3: LCM of 24 and 36

#### Step 1: Prime Factorization
- 24:
$$
24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1
$$

- 36:
$$
36 = 2 \times 18 = 2 \times 2 \times 9 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2
$$

#### Step 2: Highest powers:
- $2^3$ (from 24)
- $3^2$ (from 36)

#### Step 3: Multiply:
$$
LCM = 2^3 \times 3^2 = 8 \times 9 = 72
$$

✔ Answer: LCM = 72

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Problem 4: LCM of 24 and 56

#### Step 1: Prime Factorization
- 24:
$$
24 = 2^3 \times 3^1 \quad \text{(same as above)}
$$

- 56:
$$
56 = 2 \times 28 = 2 \times 2 \times 14 = 2 \times 2 \times 2 \times 7 = 2^3 \times 7^1
$$

#### Step 2: Highest powers:
- $2^3$ (common)
- $3^1$ (from 24)
- $7^1$ (from 56)

#### Step 3: Multiply:
$$
LCM = 2^3 \times 3^1 \times 7^1 = 8 \times 3 \times 7 = 168
$$

✔ Answer: LCM = 168

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✔ Final Answers:

| Problem | Numbers | LCM |
|--------|--------|-----|
| 1 | 50, 75 | 150 |
| 2 | 40, 60 | 120 |
| 3 | 24, 36 | 72 |
| 4 | 24, 56 | 168 |

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📝 How to Fill Out the Worksheet:

For each problem, fill in the factor trees and then write:

#### Example for 50 and 75:
```
50 = 2 × 5²
75 = 3 × 5²
LCM = 2 × 3 × 5² = 150
```

Similarly for others.

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Let me know if you'd like a filled-in version of the worksheet!