Math worksheet for practicing prime factorization to determine LCM and GCF of given number sets.

To solve this worksheet, we need to find the Least Common Multiple (LCM) and Greatest Common Factor (GCF) of each set of three numbers using prime factorization.

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🔢 How to use Prime Factorization:

- GCF: Take the *lowest power* of *common prime factors* across all numbers.
- LCM: Take the *highest power* of *all prime factors* that appear in any of the numbers.

Let’s go one by one.

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## ✔ 1) 4, 5, 24

Prime factorizations:
- 4 = 2²
- 5 = 5
- 24 = 2³ × 3

GCF: Only common prime factor? None shared by all three → GCF = 1

LCM: Highest powers: 2³, 3, 5 → 8 × 3 × 5 = 120

✔ LCM = 120, GCF = 1

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## ✔ 2) 4, 15, 16

Prime factorizations:
- 4 = 2²
- 15 = 3 × 5
- 16 = 2⁴

GCF: No prime factor common to all three → GCF = 1

LCM: 2⁴, 3, 5 → 16 × 3 × 5 = 240

✔ LCM = 240, GCF = 1

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## ✔ 3) 32, 34, 36

Prime factorizations:
- 32 = 2⁵
- 34 = 2 × 17
- 36 = 2² × 3²

GCF: Only common prime factor is 2. Lowest power = 2¹ → GCF = 2

LCM: 2⁵, 3², 17 → 32 × 9 × 17 = 288 × 17 = 4896

✔ LCM = 4896, GCF = 2

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## ✔ 4) 34, 35, 32

Prime factorizations:
- 34 = 2 × 17
- 35 = 5 × 7
- 32 = 2⁵

GCF: No prime common to all → GCF = 1

LCM: 2⁵, 5, 7, 17 → 32 × 5 × 7 × 17
= 32 × 5 = 160; 160 × 7 = 1120; 1120 × 17 = 19040

✔ LCM = 19040, GCF = 1

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## ✔ 5) 21, 24, 36

Prime factorizations:
- 21 = 3 × 7
- 24 = 2³ × 3
- 36 = 2² × 3²

GCF: Common prime = 3. Lowest power = 3¹ → GCF = 3

LCM: 2³, 3², 7 → 8 × 9 × 7 = 72 × 7 = 504

✔ LCM = 504, GCF = 3

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## ✔ 6) 10, 16, 21

Prime factorizations:
- 10 = 2 × 5
- 16 = 2⁴
- 21 = 3 × 7

GCF: No prime common to all → GCF = 1

LCM: 2⁴, 3, 5, 7 → 16 × 3 × 5 × 7 = 16 × 105 = 1680

✔ LCM = 1680, GCF = 1

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## ✔ 7) 21, 36, 32

Prime factorizations:
- 21 = 3 × 7
- 36 = 2² × 3²
- 32 = 2⁵

GCF: No prime common to all → GCF = 1

LCM: 2⁵, 3², 7 → 32 × 9 × 7 = 288 × 7 = 2016

✔ LCM = 2016, GCF = 1

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## ✔ 8) 34, 21, 14

Prime factorizations:
- 34 = 2 × 17
- 21 = 3 × 7
- 14 = 2 × 7

GCF: Common prime? 2 appears in 34 & 14, but not 21. 7 appears in 21 & 14, but not 34. No prime in all three → GCF = 1

LCM: 2, 3, 7, 17 → 2 × 3 × 7 × 17 = 6 × 7 = 42; 42 × 17 = 714

✔ LCM = 714, GCF = 1

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## ✔ 9) 35, 15, 12

Prime factorizations:
- 35 = 5 × 7
- 15 = 3 × 5
- 12 = 2² × 3

GCF: No prime common to all three → GCF = 1

LCM: 2², 3, 5, 7 → 4 × 3 × 5 × 7 = 12 × 35 = 420

✔ LCM = 420, GCF = 1

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## ✔ 10) 34, 15, 4

Prime factorizations:
- 34 = 2 × 17
- 15 = 3 × 5
- 4 = 2²

GCF: Only 2 is common between 34 and 4, but not 15 → GCF = 1

LCM: 2², 3, 5, 17 → 4 × 3 × 5 × 17 = 60 × 17 = 1020

✔ LCM = 1020, GCF = 1

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## 📝 Final Answers:

| Problem | LCM | GCF |
|---------|--------|-----|
| 1 | 120 | 1 |
| 2 | 240 | 1 |
| 3 | 4896 | 2 |
| 4 | 19040 | 1 |
| 5 | 504 | 3 |
| 6 | 1680 | 1 |
| 7 | 2016 | 1 |
| 8 | 714 | 1 |
| 9 | 420 | 1 |
| 10 | 1020 | 1 |

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You can now fill these into your worksheet! Let me know if you’d like a step-by-step breakdown for any specific problem. 😊