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 = 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 is not

Break down 9:
- 9 = 3 × 3 → both are prime

So the full tree:
```
36
/ \
2 18
/ \
2 9
/ \
3 3
```

Prime factors: 2, 2, 3, 3

→ 36 = 2 × 2 × 3 × 3 = 2² × 3²

✔ Answer:
36 = 2² × 3²

---

🔹 2. 54

Start with 54:

- 54 = 2 × 27 → 2 is prime
- 27 = 3 × 9 → 3 is prime
- 9 = 3 × 3 → both prime

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

Prime factors: 2, 3, 3, 3

→ 54 = 2 × 3³

✔ Answer:
54 = 2 × 3³

---

🔹 3. 135

Start with 135:

- 135 is odd → not divisible by 2
- Try 3: 1 + 3 + 5 = 9 → divisible by 3
So: 135 ÷ 3 = 45 → 135 = 3 × 45

Now 45 = 5 × 9 → 5 is prime, 9 = 3 × 3

Tree:
```
135
/ \
3 45
/ \
5 9
/ \
3 3
```

Prime factors: 3, 5, 3, 3 → 3³ × 5

→ 135 = 3 × 3 × 3 × 5 = 3³ × 5

✔ Answer:
135 = 3³ × 5

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🔹 4. 525

Start with 525:

- Ends in 5 → divisible by 5
525 ÷ 5 = 105 → 525 = 5 × 105

105 ÷ 5 = 21 → 105 = 5 × 21

21 = 3 × 7 → both prime

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

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

→ 525 = 5² × 3 × 7

✔ Answer:
525 = 3 × 5² × 7

(We usually write in ascending order: 3 × 5² × 7)

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🔹 5. 693

Start with 693:

- Sum of digits: 6+9+3 = 18 → divisible by 3
693 ÷ 3 = 231 → 693 = 3 × 231

231 ÷ 3 = 77 → 231 = 3 × 77

77 = 7 × 11 → both prime

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

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

→ 693 = 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 |

---

🌲 How to Fill in the Factor Trees:

#### For 36:
```
36
/ \
2 18
/ \
2 9
/ \
3 3
```
- Blue circles: 2, 2, 3, 3
- Orange boxes: 18, 9

→ 36 = 2² × 3²

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

#### For 135:
```
135
/ \
3 45
/ \
5 9
/ \
3 3
```
→ 135 = 3³ × 5

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

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

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✔ Summary:

You now have all the factor trees filled out correctly and the prime factorizations written in exponential form as requested.

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