Prime Factorization with Factor Trees Worksheet | 6th Grade PDF ... - Free Printable

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 = 2 × 18 → 2 is prime ✔
- 18 = 2 × 9 → 2 is prime ✔
- 9 = 3 × 3 → both 3s are prime ✔

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

All prime factors: 2, 2, 3, 3
→ $ 36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2 $

✔ 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 3s are prime ✔

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

Prime factors: 2, 3, 3, 3
→ $ 54 = 2 \times 3 \times 3 \times 3 = 2 \times 3^3 $

✔ 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 → both 3s are prime ✔

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

Prime factors: 5, 3, 3, 3
→ $ 135 = 5 \times 3 \times 3 \times 3 = 5 \times 3^3 $

✔ 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 are prime ✔

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

Prime factors: 5, 5, 3, 7
→ $ 525 = 5 \times 5 \times 3 \times 7 = 3 \times 5^2 \times 7 $

✔ Answer:
525 = 3 × 5² × 7

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🔹 5. 693

Start with 693.

- 693 ÷ 3 = 231 → so 693 = 3 × 231 → 3 is prime ✔
- 231 ÷ 3 = 77 → so 231 = 3 × 77 → 3 is prime ✔
- 77 = 7 × 11 → both are prime ✔

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

Prime factors: 3, 3, 7, 11
→ $ 693 = 3 \times 3 \times 7 \times 11 = 3^2 \times 7 \times 11 $

✔ Answer:
693 = 3² × 7 × 11

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## ✔ Final Answers:

| Number | Prime Factorization (Product) | Exponential Form |
|--------|-------------------------------|------------------|
| 36 | 2 × 2 × 3 × 3 | $ 2^2 \times 3^2 $ |
| 54 | 2 × 3 × 3 × 3 | $ 2 \times 3^3 $ |
| 135 | 3 × 3 × 3 × 5 | $ 3^3 \times 5 $ |
| 525 | 3 × 5 × 5 × 7 | $ 3 \times 5^2 \times 7 $ |
| 693 | 3 × 3 × 7 × 11 | $ 3^2 \times 7 \times 11 $ |

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🌲 How to Fill the Factor Trees (Visual Guide):

For each tree:
- Blue circles: prime factors
- Orange rectangles: non-prime factors (to be broken down)

#### Example: 36
```
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 pattern for others.

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Let me know if you'd like a printable version or help filling out the worksheet visually!