Math worksheet for practicing prime factorization to determine LCM and GCF.

Let's solve each problem step by step using prime factorization to find the LCM (Least Common Multiple) and GCF (Greatest Common Factor).

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🔷 Key Concepts:

- Prime Factorization: Expressing a number as a product of prime numbers.
- GCF: Multiply the common prime factors with the lowest powers.
- LCM: Multiply all prime factors present in any number, with the highest powers.

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We’ll go through each problem:

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1) 4, 5, 24

Prime Factorization:
- 4 = 2²
- 5 = 5¹
- 24 = 2³ × 3¹

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

LCM: Take highest powers:
- 2³, 3¹, 5¹ → 8 × 3 × 5 = 120

✔ LCM = 120, GCF = 1

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2) 4, 15, 16

Prime Factorization:
- 4 = 2²
- 15 = 3¹ × 5¹
- 16 = 2⁴

GCF: No common prime across all → GCF = 1

LCM: Highest powers:
- 2⁴, 3¹, 5¹ → 16 × 3 × 5 = 240

✔ LCM = 240, GCF = 1

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3) 32, 34, 36

Prime Factorization:
- 32 = 2⁵
- 34 = 2¹ × 17¹
- 36 = 2² × 3²

GCF: Only common prime is 2 → lowest power = 2¹ → GCF = 2

LCM: Highest powers:
- 2⁵, 3², 17¹ → 32 × 9 × 17 = 32 × 153 = 4896

✔ LCM = 4896, GCF = 2

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4) 34, 35, 32

Prime Factorization:
- 34 = 2¹ × 17¹
- 35 = 5¹ × 7¹
- 32 = 2⁵

GCF: No common prime across all → GCF = 1

LCM: All primes:
- 2⁵, 5¹, 7¹, 17¹ → 32 × 5 × 7 × 17 = 32 × 595 = 19,040

✔ LCM = 19,040, GCF = 1

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5) 21, 24, 36

Prime Factorization:
- 21 = 3¹ × 7¹
- 24 = 2³ × 3¹
- 36 = 2² × 3²

GCF: Common prime = 3 → lowest power = 3¹ → GCF = 3

LCM: Highest powers:
- 2³, 3², 7¹ → 8 × 9 × 7 = 504

✔ LCM = 504, GCF = 3

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6) 10, 16, 21

Prime Factorization:
- 10 = 2¹ × 5¹
- 16 = 2⁴
- 21 = 3¹ × 7¹

GCF: No common prime → GCF = 1

LCM: All primes:
- 2⁴, 3¹, 5¹, 7¹ → 16 × 3 × 5 × 7 = 16 × 105 = 1,680

✔ LCM = 1,680, GCF = 1

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7) 21, 36, 32

Prime Factorization:
- 21 = 3¹ × 7¹
- 36 = 2² × 3²
- 32 = 2⁵

GCF: Only common prime? 3 appears in 21 and 36 → but not in 32 → no common across all → GCF = 1

Wait: check again — only 3 is shared between 21 and 36, but 32 has no 3. So no common prime across all three → GCF = 1

LCM: Highest powers:
- 2⁵, 3², 7¹ → 32 × 9 × 7 = 32 × 63 = 2,016

✔ LCM = 2,016, GCF = 1

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8) 34, 21, 14

Prime Factorization:
- 34 = 2¹ × 17¹
- 21 = 3¹ × 7¹
- 14 = 2¹ × 7¹

GCF: Look for common primes across all:
- 2: in 34 and 14, not 21 → no
- 7: in 21 and 14, not 34 → no
- 3: only in 21 → no
- 17: only in 34 → no
→ No common prime → GCF = 1

LCM: All primes:
- 2¹, 3¹, 7¹, 17¹ → 2 × 3 × 7 × 17 = 6 × 119 = 714

✔ LCM = 714, GCF = 1

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9) 35, 15, 12

Prime Factorization:
- 35 = 5¹ × 7¹
- 15 = 3¹ × 5¹
- 12 = 2² × 3¹

GCF: Common prime?
- 5: in 35 and 15 → not in 12 → no
- 3: in 15 and 12 → not in 35 → no
→ No common across all → GCF = 1

LCM: Highest powers:
- 2², 3¹, 5¹, 7¹ → 4 × 3 × 5 × 7 = 4 × 105 = 420

✔ LCM = 420, GCF = 1

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10) 34, 15, 4

Prime Factorization:
- 34 = 2¹ × 17¹
- 15 = 3¹ × 5¹
- 4 = 2²

GCF: Check common primes:
- 2: in 34 and 4 → but not in 15 → no
- Others: unique → no common prime → GCF = 1

LCM: Highest powers:
- 2², 3¹, 5¹, 17¹ → 4 × 3 × 5 × 17 = 4 × 255 = 1,020

✔ LCM = 1,020, GCF = 1

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✔ Final Answers:

| Problem | LCM | GCF |
|--------|---------|-----|
| 1) | 120 | 1 |
| 2) | 240 | 1 |
| 3) | 4896 | 2 |
| 4) | 19,040 | 1 |
| 5) | 504 | 3 |
| 6) | 1,680 | 1 |
| 7) | 2,016 | 1 |
| 8) | 714 | 1 |
| 9) | 420 | 1 |
| 10) | 1,020 | 1 |

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Let me know if you'd like this formatted as a printable answer sheet!