Primes and Composites Worksheets | Printable PDF Worksheets - Free Printable

Let's solve each of the prime factor trees step by step. The goal is to break down each number into its prime factors using a factor tree, where:

- Blue circles = Prime factors
- Yellow rectangles = Not prime (composite) factors

We’ll go through each number and fill in the factor trees and write the final prime factorization.

---

🔹 1. 16

Start with 16.

- 16 ÷ 2 = 8 → 2 is prime ✔
- 8 ÷ 2 = 4 → 2 is prime ✔
- 4 ÷ 2 = 2 → 2 is prime ✔
- 2 ÷ 2 = 1 → stop

So:
```
16
/ \
2 8
/ \
2 4
/ \
2 2
```

But since we need to use the diagram format:

- Top: 16
- Split into: 2 (blue) and 8 (yellow)
- 8 splits into: 2 (blue) and 4 (yellow)
- 4 splits into: 2 (blue) and 2 (blue)

✔ So the tree is:
```
16
/ \
2 8
/ \
2 4
/ \
2 2
```

Prime factorization:
16 = 2 × 2 × 2 × 2 = 2⁴

> ✔ Answer: 16 = 2⁴

---

🔹 2. 36

Break down 36:

- 36 ÷ 2 = 18 → 2 is prime ✔
- 18 ÷ 2 = 9 → 2 is prime ✔
- 9 ÷ 3 = 3 → 3 is prime ✔
- 3 ÷ 3 = 1 → stop

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

Fill in:
- 36 → 2 (blue), 18 (yellow)
- 18 → 2 (blue), 9 (yellow)
- 9 → 3 (blue), 3 (blue)

✔ So:
36 = 2 × 2 × 3 × 3 = 2² × 3²

> ✔ Answer: 36 = 2² × 3²

---

🔹 3. 54

Break down 54:

- 54 ÷ 2 = 27 → 2 is prime ✔
- 27 ÷ 3 = 9 → 3 is prime ✔
- 9 ÷ 3 = 3 → 3 is prime ✔
- 3 ÷ 3 = 1 → stop

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

Fill in:
- 54 → 2 (blue), 27 (yellow)
- 27 → 3 (blue), 9 (yellow)
- 9 → 3 (blue), 3 (blue)

✔ So:
54 = 2 × 3 × 3 × 3 = 2 × 3³

> ✔ Answer: 54 = 2 × 3³

---

🔹 4. 135

Break down 135:

- 135 ÷ 3 = 45 → 3 is prime ✔
- 45 ÷ 3 = 15 → 3 is prime ✔
- 15 ÷ 3 = 5 → 3 is prime ✔
- 5 ÷ 5 = 1 → 5 is prime ✔

Tree:
```
135
/ \
3 45
/ \
3 15
/ \
3 5
```

Fill in:
- 135 → 3 (blue), 45 (yellow)
- 45 → 3 (blue), 15 (yellow)
- 15 → 3 (blue), 5 (blue)

✔ So:
135 = 3 × 3 × 3 × 5 = 3³ × 5

> ✔ Answer: 135 = 3³ × 5

---

🔹 5. 525

Break down 525:

- 525 ÷ 5 = 105 → 5 is prime ✔
- 105 ÷ 5 = 21 → 5 is prime ✔
- 21 ÷ 3 = 7 → 3 is prime ✔
- 7 ÷ 7 = 1 → 7 is prime ✔

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

Fill in:
- 525 → 5 (blue), 105 (yellow)
- 105 → 5 (blue), 21 (yellow)
- 21 → 3 (blue), 7 (blue)

✔ So:
525 = 5 × 5 × 3 × 7 = 3 × 5² × 7

> ✔ Answer: 525 = 3 × 5² × 7

---

🔹 6. 693

Break down 693:

- 693 ÷ 3 = 231 → 3 is prime ✔
- 231 ÷ 3 = 77 → 3 is prime ✔
- 77 ÷ 7 = 11 → 7 is prime ✔
- 11 ÷ 11 = 1 → 11 is prime ✔

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

Fill in:
- 693 → 3 (blue), 231 (yellow)
- 231 → 3 (blue), 77 (yellow)
- 77 → 7 (blue), 11 (blue)

✔ So:
693 = 3 × 3 × 7 × 11 = 3² × 7 × 11

> ✔ Answer: 693 = 3² × 7 × 11

---

✔ Final Answers:

| Number | Prime Factorization |
|--------|---------------------|
| 16 | 2⁴ |
| 36 | 2² × 3² |
| 54 | 2 × 3³ |
| 135 | 3³ × 5 |
| 525 | 3 × 5² × 7 |
| 693 | 3² × 7 × 11 |

---

📝 How to Fill in the Diagrams:

For each tree:

#### Example: 16
```
16
/ \
[2] [8]
/ \
[2] [4]
/ \
[2] [2]
```
- Blue circles: 2, 2, 2, 2
- Yellow rectangles: 8, 4

Similarly for others.

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✔ Summary:

You now have all the prime factorizations and can fill in the trees accordingly. The key idea is:

- Always divide by the smallest prime possible.
- Continue until all branches end in prime numbers (blue).
- Composite numbers (non-prime) go in yellow rectangles.

Let me know if you'd like this as a printable answer sheet!