Prime Factor Decomposition Using Factor Trees Worksheet ... - Free Printable

Let's solve each of the factor trees in Section A step by step, finding the prime factor decomposition of each number and expressing the answer in index notation.

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🔍 Understanding the Key

- Blue circles (🔵) = Prime factors
- Orange squares (🟨) = Not prime factors (composite numbers or non-primes)

We break down each number into its prime factors using factor trees. We continue breaking down non-prime factors until all branches end in prime numbers.

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## ✔ 1. 36

Start with 36:

- 36 = 2 × 18 → 2 is prime (🔵), 18 is not (🟨)
- 18 = 2 × 9 → 2 is prime (🔵), 9 is not (🟨)
- 9 = 3 × 3 → both are prime (🔵)

So:
```
36
/ \
2 18
/ \
2 9
/ \
3 3
```

Prime factors: 2, 2, 3, 3
→ 36 = 2² × 3²

✔ Answer:
36 = 2² × 3²

---

## ✔ 2. 54

Start with 54:

- 54 = 2 × 27 → 2 is prime (🔵), 27 is not (🟨)
- 27 = 3 × 9 → 3 is prime (🔵), 9 is not (🟨)
- 9 = 3 × 3 → both prime (🔵)

```
54
/ \
2 27
/ \
3 9
/ \
3 3
```

Prime factors: 2, 3, 3, 3
→ 54 = 2 × 3³

✔ Answer:
54 = 2 × 3³

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## ✔ 3. 135

Start with 135:

- 135 = 5 × 27 → 5 is prime (🔵), 27 is not (🟨)
- 27 = 3 × 9 → 3 is prime (🔵), 9 is not (🟨)
- 9 = 3 × 3 → both prime (🔵)

```
135
/ \
5 27
/ \
3 9
/ \
3 3
```

Prime factors: 5, 3, 3, 3
→ 135 = 3³ × 5

✔ Answer:
135 = 3³ × 5

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## ✔ 4. 525

Start with 525:

- 525 = 5 × 105 → 5 is prime (🔵), 105 is not (🟨)
- 105 = 5 × 21 → 5 is prime (🔵), 21 is not (🟨)
- 21 = 3 × 7 → both prime (🔵)

```
525
/ \
5 105
/ \
5 21
/ \
3 7
```

Prime factors: 5, 5, 3, 7
→ 525 = 3 × 5² × 7

✔ Answer:
525 = 3 × 5² × 7

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## ✔ 5. 693

Start with 693:

- 693 = 3 × 231 → 3 is prime (🔵), 231 is not (🟨)
- 231 = 3 × 77 → 3 is prime (🔵), 77 is not (🟨)
- 77 = 7 × 11 → both prime (🔵)

```
693
/ \
3 231
/ \
3 77
/ \
7 11
```

Prime factors: 3, 3, 7, 11
→ 693 = 3² × 7 × 11

✔ Answer:
693 = 3² × 7 × 11

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📌 Final Answers (in index notation):

| Number | Prime Factor Decomposition |
|--------|---------------------------|
| 36 | 2² × 3² |
| 54 | 2 × 3³ |
| 135 | 3³ × 5 |
| 525 | 3 × 5² × 7 |
| 693 | 3² × 7 × 11 |

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✔ Summary of Completed Factor Trees:

#### 36:
```
36
/ \
2 18
/ \
2 9
/ \
3 3
```
→ 36 = 2² × 3²

#### 54:
```
54
/ \
2 27
/ \
3 9
/ \
3 3
```
→ 54 = 2 × 3³

#### 135:
```
135
/ \
5 27
/ \
3 9
/ \
3 3
```
→ 135 = 3³ × 5

#### 525:
```
525
/ \
5 105
/ \
5 21
/ \
3 7
```
→ 525 = 3 × 5² × 7

#### 693:
```
693
/ \
3 231
/ \
3 77
/ \
7 11
```
→ 693 = 3² × 7 × 11

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Let me know if you'd like this as a printable version or need help with any other math topic! 😊