Relationship between GCF and LCM | Worksheets | Math Center - Free Printable

Let’s solve this step by step.

We are given pairs of numbers and need to fill in four columns for each pair:
- Multiplication → just multiply the two numbers.
- GCF → Greatest Common Factor (the largest number that divides both).
- LCM → Least Common Multiple (the smallest number that both divide into).
- Then we’ll check if LCM = Multiplication / GCF is true.

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Row 1: 6, 7

- Multiplication: 6 × 7 = 42
- GCF: Factors of 6: 1,2,3,6; Factors of 7: 1,7 → only common factor is 1
- LCM: Since they’re coprime (GCF=1), LCM = 6×7 = 42

Check: 42 ÷ 1 = 42 → matches LCM ✔

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Row 2: 8, 9

- Multiplication: 8 × 9 = 72
- GCF: 8 = 2³, 9 = 3² → no common factors → 1
- LCM: 8×9 = 72

Check: 72 ÷ 1 = 72 → matches ✔

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Row 3: 4, 9

- Multiplication: 4 × 9 = 36
- GCF: 4 = 2², 9 = 3² → no common factors → 1
- LCM: 4×9 = 36

Check: 36 ÷ 1 = 36 → matches ✔

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Row 4: 5, 7

- Multiplication: 5 × 7 = 35
- GCF: both prime → 1
- LCM: 5×7 = 35

Check: 35 ÷ 1 = 35 → matches ✔

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Row 5: 6, 12

- Multiplication: 6 × 12 = 72
- GCF: 6 divides 12 → so GCF = 6
- LCM: since 12 is multiple of 6, LCM = 12

Check: 72 ÷ 6 = 12 → matches ✔

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Row 6: 4, 16

- Multiplication: 4 × 16 = 64
- GCF: 4 divides 16 → GCF = 4
- LCM: 16 is multiple of 4 → LCM = 16

Check: 64 ÷ 4 = 16 → matches ✔

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Row 7: 10, 40

- Multiplication: 10 × 40 = 400
- GCF: 10 divides 40 → GCF = 10
- LCM: 40 is multiple of 10 → LCM = 40

Check: 400 ÷ 10 = 40 → matches ✔

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Row 8: 7, 14

- Multiplication: 7 × 14 = 98
- GCF: 7 divides 14 → GCF = 7
- LCM: 14 is multiple of 7 → LCM = 14

Check: 98 ÷ 7 = 14 → matches ✔

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Row 9: 6, 8

- Multiplication: 6 × 8 = 48
- GCF: factors of 6: 1,2,3,6; factors of 8: 1,2,4,8 → common: 1,2 → GCF = 2
- LCM: list multiples of 6: 6,12,18,24,30...; multiples of 8: 8,16,24,... → first common is 24

Check: 48 ÷ 2 = 24 → matches ✔

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Row 10: 8, 12

- Multiplication: 8 × 12 = 96
- GCF: 8 = 2³, 12 = 2²×3 → common: 2² = 4
- LCM: use formula or list: multiples of 8: 8,16,24,32...; 12: 12,24,... → LCM = 24

Check: 96 ÷ 4 = 24 → matches ✔

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Row 11: 10, 15

- Multiplication: 10 × 15 = 150
- GCF: 10 = 2×5, 15 = 3×5 → common: 5
- LCM: multiples of 10: 10,20,30,40...; 15: 15,30,... → LCM = 30

Check: 150 ÷ 5 = 30 → matches ✔

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Row 12: 12, 18

- Multiplication: 12 × 18 = 216
- GCF: 12 = 2²×3, 18 = 2×3² → common: 2×3 = 6
- LCM: multiples of 12: 12,24,36,48...; 18: 18,36,... → LCM = 36

Check: 216 ÷ 6 = 36 → matches ✔

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Now let’s put all answers in table form:

| The numbers | Multiplication | GCF | LCM |
|-------------|----------------|-----|-----|
| 6, 7 | 42 | 1 | 42 |
| 8, 9 | 72 | 1 | 72 |
| 4, 9 | 36 | 1 | 36 |
| 5, 7 | 35 | 1 | 35 |
| 6, 12 | 72 | 6 | 12 |
| 4, 16 | 64 | 4 | 16 |
| 10, 40 | 400 | 10 | 40 |
| 7, 14 | 98 | 7 | 14 |
| 6, 8 | 48 | 2 | 24 |
| 8, 12 | 96 | 4 | 24 |
| 10, 15 | 150 | 5 | 30 |
| 12, 18 | 216 | 6 | 36 |

And yes — for every row, LCM = Multiplication ÷ GCF holds true.

Final Answer:
The completed table is as shown above, and it is correct to say:
The LCM = Multiplication / GCF