Least Common Multiples with factor trees worksheet - Free Printable

Let's solve each part of the Least Common Multiple (LCM) worksheet step by step, following the instructions:

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🔷 Goal:

Find the Least Common Multiple (LCM) using prime factorization:
1. Make factor trees to break down numbers into prime factors.
2. Write the prime factorization of each number.
3. Multiply the highest powers of all prime factors to get the LCM.

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## ✔ Part a: 9 & 20 (Already solved as example)

- Prime factorization of 9: $3 \times 3 = 3^2$
- Prime factorization of 20: $5 \times 2 \times 2 = 2^2 \times 5$
- LCM: Take highest powers: $2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$

✔ LCM = 180

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## 🟦 Part b: 2 & 24

Step 1: Factor Trees

#### For 2:
- 2 is already a prime number, so its factor tree stops here.

```
2
/
(no further branches)
```

So:
- Prime factorization of 2: $2$

#### For 24:
Break it down:

```
24
/ \
6 4
/ \ / \
2 3 2 2
```

So:
- Prime factorization of 24: $2 \times 2 \times 2 \times 3 = 2^3 \times 3$

Step 2: Find LCM

List all primes with maximum exponents:
- $2^3$ (from 24) — higher than $2^1$ in 2
- $3^1$ (only from 24)

Multiply:
$$
2^3 \times 3 = 8 \times 3 = 24
$$

✔ LCM = 24

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Fill in the blanks for part b:

- Prime factorization 2: $2$
- Prime factorization 24: $2 \times 2 \times 2 \times 3$ or $2^3 \times 3$
- Multiply to find LCM: $2^3 \times 3 = 8 \times 3 = 24$
- LCM: $24$

---

## 🟨 Part c: 10 & 30

Step 1: Factor Trees

#### For 10:
```
10
/ \
2 5
```
- Prime factorization: $2 \times 5$

#### For 30:
```
30
/ \
6 5
/ \
2 3
```
- So: $2 \times 3 \times 5$

Or more clearly:
```
30
/ \
5 6
/ \
2 3
```

So:
- Prime factorization of 30: $2 \times 3 \times 5$

Step 2: LCM

Take highest powers of all primes:
- $2^1$: appears in both
- $3^1$: only in 30
- $5^1$: in both

So:
$$
2 \times 3 \times 5 = 30
$$

✔ LCM = 30

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Fill in the blanks for part c:

- Prime factorization 10: $2 \times 5$
- Prime factorization 30: $2 \times 3 \times 5$
- Multiply to find LCM: $2 \times 3 \times 5 = 30$
- LCM: $30$

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## 🟪 Part d: 12 & 18

Step 1: Factor Trees

#### For 12:
```
12
/ \
6 2
/ \
2 3
```
So: $2 \times 2 \times 3 = 2^2 \times 3$

#### For 18:
```
18
/ \
9 2
/ \
3 3
```
So: $3 \times 3 \times 2 = 2 \times 3^2$

Step 2: LCM

Primes involved: 2 and 3

- Highest power of 2: $2^2$ (from 12)
- Highest power of 3: $3^2$ (from 18)

Multiply:
$$
2^2 \times 3^2 = 4 \times 9 = 36
$$

✔ LCM = 36

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Fill in the blanks for part d:

- Prime factorization 12: $2 \times 2 \times 3$ or $2^2 \times 3$
- Prime factorization 18: $2 \times 3 \times 3$ or $2 \times 3^2$
- Multiply to find LCM: $2^2 \times 3^2 = 4 \times 9 = 36$
- LCM: $36$

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## ✔ Final Answers Summary:

| Problem | LCM |
|--------|-----|
| a. 9 & 20 | 180 |
| b. 2 & 24 | 24 |
| c. 10 & 30 | 30 |
| d. 12 & 18 | 36 |

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## 🔍 Key Reminder:
> We don’t multiply all prime factors together blindly. We take each prime factor the maximum number of times it appears in either number.

For example:
- In 12 = $2^2 \times 3$, and 18 = $2 \times 3^2$
- So LCM uses $2^2$ and $3^2$

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Let me know if you'd like this formatted as a completed worksheet!