Worksheet for finding prime factorization using factor trees, including examples and exercises for numbers 16, 36, 54, 135, 525, and 693.

Let's solve each problem step by step using factor trees to find the prime factorization of each number. We’ll follow the example given for 16, where we break down the number into factors until all are prime numbers (circled in blue), and non-prime factors (in orange) are broken down further.

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🔹 1. 36

We start with 36.

- 36 = 6 × 6 → but better to use smaller factors.
- Let’s split 36 as:
36 = 2 × 18 (2 is prime, so circle it)

Now break down 18:
- 18 = 2 × 9 (2 is prime)
- 9 = 3 × 3 (both 3s are prime)

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

Prime factors: 2, 2, 3, 3

→ 36 = 2 × 2 × 3 × 3 = 2² × 3²

✔ Answer: 36 = 2² × 3²

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🔹 2. 54

Start with 54:

- 54 = 2 × 27 (2 is prime)
- 27 = 3 × 9 (3 is prime)
- 9 = 3 × 3 (both prime)

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

Prime factors: 2, 3, 3, 3

→ 54 = 2 × 3 × 3 × 3 = 2 × 3³

✔ Answer: 54 = 2 × 3³

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🔹 3. 135

Start with 135:

- 135 = 5 × 27 (5 is prime)
- 27 = 3 × 9 (3 is prime)
- 9 = 3 × 3

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

Prime factors: 5, 3, 3, 3

→ 135 = 5 × 3 × 3 × 3 = 5 × 3³

✔ Answer: 135 = 3³ × 5

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🔹 4. 525

Start with 525:

- 525 = 5 × 105 (5 is prime)
- 105 = 5 × 21 (5 is prime)
- 21 = 3 × 7 (both prime)

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

Prime factors: 5, 5, 3, 7

→ 525 = 5 × 5 × 3 × 7 = 3 × 5² × 7

✔ Answer: 525 = 3 × 5² × 7

---

🔹 5. 693

Start with 693:

- Check divisibility:
Sum of digits: 6+9+3 = 18 → divisible by 3 → yes.

- 693 ÷ 3 = 231 → so 693 = 3 × 231
- 231 ÷ 3 = 77 → 231 = 3 × 77
- 77 = 7 × 11 (both prime)

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

Prime factors: 3, 3, 7, 11

→ 693 = 3 × 3 × 7 × 11 = 3² × 7 × 11

✔ Answer: 693 = 3² × 7 × 11

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✔ Final Answers:

| Number | Prime Factorization (Product) | Exponential Form |
|--------|-------------------------------|------------------|
| 36 | 2 × 2 × 3 × 3 | 2² × 3² |
| 54 | 2 × 3 × 3 × 3 | 2 × 3³ |
| 135 | 3 × 3 × 3 × 5 | 3³ × 5 |
| 525 | 3 × 5 × 5 × 7 | 3 × 5² × 7 |
| 693 | 3 × 3 × 7 × 11 | 3² × 7 × 11 |

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🌲 How to Fill in the Trees:

Use the key:
- Blue circles: Prime factors
- Orange rectangles: Not prime (need to be broken down)

#### Example: 36 Tree

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

Fill in:
- First level: 2 (blue), 18 (orange)
- Second: 2 (blue), 9 (orange)
- Third: 3 (blue), 3 (blue)

Then write: 36 = 2² × 3²

Repeat this process for others.

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Let me know if you'd like a printable version or a visual diagram!