Gcf and Lcm Worksheet Elegant Word Problems Involving Gcf and Lcm ... - Free Printable

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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🔷 Rules:

- GCF: Multiply the common prime factors with the lowest powers.
- LCM: Multiply all prime factors (common and unique) with the highest powers.

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We’ll go through each set of numbers:

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1) 4, 5, 24

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

GCF: No common prime factors in 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

Factorizations:
- 4 = 2²
- 15 = 3¹ × 5¹
- 16 = 2⁴

GCF: No common prime factor 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

Factorizations:
- 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 × 9 = 288; 288 × 17 = 4896

✔ LCM = 4896, GCF = 2

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4) 34, 35, 32

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

GCF: No common prime across all → GCF = 1

LCM: Highest powers:
- 2⁵, 5¹, 7¹, 17¹ → 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

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

GCF: Only common factor is 3 → 3¹ → GCF = 3

LCM: Highest powers:
- 2³, 3², 7¹ → 8 × 9 × 7 = 72 × 7 = 504

✔ LCM = 504, GCF = 3

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6) 10, 16, 21

Factorizations:
- 10 = 2¹ × 5¹
- 16 = 2⁴
- 21 = 3¹ × 7¹

GCF: No common prime across all → GCF = 1

LCM: Highest powers:
- 2⁴, 3¹, 5¹, 7¹ → 16 × 3 × 5 × 7 =
16 × 3 = 48; 48 × 5 = 240; 240 × 7 = 1680

✔ LCM = 1680, GCF = 1

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7) 21, 36, 32

Factorizations:
- 21 = 3¹ × 7¹
- 36 = 2² × 3²
- 32 = 2⁵

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

LCM: Highest powers:
- 2⁵, 3², 7¹ → 32 × 9 × 7 =
32 × 9 = 288; 288 × 7 = 2016

✔ LCM = 2016, GCF = 1

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8) 34, 21, 14

Factorizations:
- 34 = 2¹ × 17¹
- 21 = 3¹ × 7¹
- 14 = 2¹ × 7¹

GCF: Look for common primes in all three:
- 2: only in 34 and 14 → not in 21 → skip
- 7: in 21 and 14 → not in 34 → skip
- 3: only in 21 → skip
- 17: only in 34 → skip

No common prime → GCF = 1

LCM: Highest powers:
- 2¹, 3¹, 7¹, 17¹ → 2 × 3 × 7 × 17 =
2 × 3 = 6; 6 × 7 = 42; 42 × 17 = 714

✔ LCM = 714, GCF = 1

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9) 35, 15, 12

Factorizations:
- 35 = 5¹ × 7¹
- 15 = 3¹ × 5¹
- 12 = 2² × 3¹

GCF: Only common prime across all? 5: in 35 and 15, but not in 12 → no. 3: in 15 and 12, not in 35 → no. So GCF = 1

LCM: Highest powers:
- 2², 3¹, 5¹, 7¹ → 4 × 3 × 5 × 7 =
4 × 3 = 12; 12 × 5 = 60; 60 × 7 = 420

✔ LCM = 420, GCF = 1

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10) 34, 15, 4

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

GCF: Check common primes:
- 2: in 34 and 4, but not in 15 → no common in all → GCF = 1

LCM: Highest powers:
- 2², 3¹, 5¹, 17¹ → 4 × 3 × 5 × 17 =
4 × 3 = 12; 12 × 5 = 60; 60 × 17 = 1020

✔ LCM = 1020, GCF = 1

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✔ Final Answers:

| # | Numbers | LCM | GCF |
|---|----------------|---------|-----|
| 1 | 4, 5, 24 | 120 | 1 |
| 2 | 4, 15, 16 | 240 | 1 |
| 3 | 32, 34, 36 | 4896 | 2 |
| 4 | 34, 35, 32 | 19040 | 1 |
| 5 | 21, 24, 36 | 504 | 3 |
| 6 | 10, 16, 21 | 1680 | 1 |
| 7 | 21, 36, 32 | 2016 | 1 |
| 8 | 34, 21, 14 | 714 | 1 |
| 9 | 35, 15, 12 | 420 | 1 |
|10 | 34, 15, 4 | 1020 | 1 |

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