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

Let's solve the problem step by step.

We are given a table with pairs of numbers. For each pair, we need to:

1. Multiply the two numbers (Multiplication)
2. Find the Greatest Common Factor (GCF) of the two numbers
3. Find the Least Common Multiple (LCM) of the two numbers

Then, at the end, we will evaluate whether the formula:

> LCM = Multiplication / GCF

is correct.

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🔍 Key Concept:

There is a mathematical relationship between two numbers, their GCF, and LCM:

> LCM(a, b) × GCF(a, b) = a × b

So,
> LCM = (a × b) / GCF

This means:
✔ Yes, it is correct to say:
The LCM = Multiplication / GCF, as long as both numbers are positive integers.

We'll verify this for each row.

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Let’s fill in the table one by one.

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1. Numbers: 6, 7

- Multiplication: 6 × 7 = 42
- GCF: 6 and 7 are coprime → GCF = 1
- LCM: Since they're coprime, LCM = 6 × 7 = 42

Check: 42 / 1 = 42 ✔

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2. Numbers: 8, 9

- Multiplication: 8 × 9 = 72
- GCF: 8 and 9 are coprime → GCF = 1
- LCM: 72

Check: 72 / 1 = 72 ✔

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3. Numbers: 4, 9

- Multiplication: 4 × 9 = 36
- GCF: 4 and 9 → no common factors other than 1 → GCF = 1
- LCM: 36

Check: 36 / 1 = 36 ✔

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4. Numbers: 5, 7

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

Check: 35 / 1 = 35 ✔

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5. Numbers: 6, 12

- Multiplication: 6 × 12 = 72
- GCF: Factors of 6: 1,2,3,6; 12: 1,2,3,4,6,12 → GCF = 6
- LCM: Use formula: LCM = 72 / 6 = 12

Alternatively: LCM of 6 and 12 is 12 (since 12 is divisible by 6)

Check: 72 / 6 = 12 ✔

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6. Numbers: 4, 16

- Multiplication: 4 × 16 = 64
- GCF: 4 divides 16 → GCF = 4
- LCM: 64 / 4 = 16

LCM(4,16) = 16 ✔

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7. Numbers: 10, 40

- Multiplication: 10 × 40 = 400
- GCF: 10 divides 40 → GCF = 10
- LCM: 400 / 10 = 40

LCM(10,40) = 40 ✔

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8. Numbers: 7, 14

- Multiplication: 7 × 14 = 98
- GCF: 7 divides 14 → GCF = 7
- LCM: 98 / 7 = 14

LCM(7,14) = 14 ✔

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9. Numbers: 6, 8

- Multiplication: 6 × 8 = 48
- GCF: 6 = 2×3, 8 = 2³ → GCF = 2
- LCM: 48 / 2 = 24

LCM(6,8): multiples of 6: 6,12,18,24,...; multiples of 8: 8,16,24 → LCM = 24 ✔

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10. Numbers: 8, 12

- Multiplication: 8 × 12 = 96
- GCF: 8 = 2³, 12 = 2²×3 → GCF = 2² = 4
- LCM: 96 / 4 = 24

LCM(8,12): 24 ✔

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11. Numbers: 10, 15

- Multiplication: 10 × 15 = 150
- GCF: 10 = 2×5, 15 = 3×5 → GCF = 5
- LCM: 150 / 5 = 30

LCM(10,15) = 30 ✔

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12. Numbers: 12, 18

- Multiplication: 12 × 18 = 216
- GCF: 12 = 2²×3, 18 = 2×3² → GCF = 2×3 = 6
- LCM: 216 / 6 = 36

LCM(12,18): multiples → 36 ✔

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## ✔ Final Table:

| 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 |

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## 🟩 Answer to the Question:

> Would it be correct to say: The LCM = Multiplication / GCF?

✔ Yes, it is correct.

Because for any two positive integers $ a $ and $ b $:

$$
\text{LCM}(a,b) \times \text{GCF}(a,b) = a \times b
$$

So,

$$
\text{LCM} = \frac{a \times b}{\text{GCF}}
$$

This is a fundamental identity in number theory.

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✔ Final Answer:

Yes, it is correct to say:
The LCM = Multiplication / GCF ✔

And here is the completed table above.