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

Let's solve each problem 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 branches end in prime numbers (circled in blue), and non-prime factors are in orange.

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🔷 Key Reminder:

- Blue circles = Prime factors
- Orange boxes = Not prime factors (composite or non-prime)

We will:
1. Break down each number into two factors.
2. Continue breaking down non-prime factors.
3. Stop when all branches are primes.
4. Write the final answer as a product of primes and in exponential form.

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## ✔ 1. 36

Start with 36:

```
36
/ \
2 18 ← 18 is not prime → break it down
/ \
2 9 ← 9 is not prime → break it down
/ \
3 3
```

So:
- 36 = 2 × 2 × 3 × 3
- = $ 2^2 \times 3^2 $

✔ Answer:
36 = 2 × 2 × 3 × 3 = $ 2^2 \times 3^2 $

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## ✔ 2. 54

Start with 54:

```
54
/ \
2 27 ← 27 is not prime → break it down
/ \
3 9 ← 9 is not prime → break it down
/ \
3 3
```

So:
- 54 = 2 × 3 × 3 × 3
- = $ 2 \times 3^3 $

✔ Answer:
54 = 2 × 3 × 3 × 3 = $ 2 \times 3^3 $

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## ✔ 3. 135

Start with 135:

```
135
/ \
5 27 ← 27 is not prime → break it down
/ \
3 9 ← 9 is not prime → break it down
/ \
3 3
```

So:
- 135 = 5 × 3 × 3 × 3
- = $ 5 \times 3^3 $

✔ Answer:
135 = 5 × 3 × 3 × 3 = $ 5 \times 3^3 $

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## ✔ 4. 525

Start with 525:

First, check divisibility:
- Ends in 5 → divisible by 5

```
525
/ \
5 105 ← 105 is not prime → divide by 5 again
/ \
5 21 ← 21 is not prime → break into 3 × 7
/ \
3 7
```

So:
- 525 = 5 × 5 × 3 × 7
- = $ 3 \times 5^2 \times 7 $

✔ Answer:
525 = 3 × 5 × 5 × 7 = $ 3 \times 5^2 \times 7 $

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## ✔ 5. 693

Now 693:

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

```
693
/ \
3 231 ← 231: sum = 2+3+1=6 → divisible by 3
/ \
3 77 ← 77 = 7 × 11
/ \
7 11
```

So:
- 693 = 3 × 3 × 7 × 11
- = $ 3^2 \times 7 \times 11 $

✔ Answer:
693 = 3 × 3 × 7 × 11 = $ 3^2 \times 7 \times 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 | 5 × 3 × 3 × 3 | $ 5 \times 3^3 $ |
| 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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✔ Fill in the Trees:

#### 36:
```
36
/ \
2 18
/ \
2 9
/ \
3 3
```
→ 36 = $ 2^2 \times 3^2 $

#### 54:
```
54
/ \
2 27
/ \
3 9
/ \
3 3
```
→ 54 = $ 2 \times 3^3 $

#### 135:
```
135
/ \
5 27
/ \
3 9
/ \
3 3
```
→ 135 = $ 5 \times 3^3 $

#### 525:
```
525
/ \
5 105
/ \
5 21
/ \
3 7
```
→ 525 = $ 3 \times 5^2 \times 7 $

#### 693:
```
693
/ \
3 231
/ \
3 77
/ \
7 11
```
→ 693 = $ 3^2 \times 7 \times 11 $

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