Worksheet for practicing prime factorization using number trees.

To solve the problem of finding the prime factors of each number using prime factorization trees, we need to break down each number into its prime factors step by step. Let's go through each number one by one.

---

1. Number: 27

- Start with 27.
- Divide by the smallest prime number, which is 3:
\[
27 \div 3 = 9
\]
- Now, factorize 9:
\[
9 \div 3 = 3
\]
- Finally, 3 is a prime number.
- The prime factorization of 27 is:
\[
27 = 3 \times 3 \times 3 = 3^3
\]

Prime Factorization Tree:
```
27
/ \
3 9
/ \
3 3
```

---

2. Number: 20

- Start with 20.
- Divide by the smallest prime number, which is 2:
\[
20 \div 2 = 10
\]
- Now, factorize 10:
\[
10 \div 2 = 5
\]
- Finally, 5 is a prime number.
- The prime factorization of 20 is:
\[
20 = 2 \times 2 \times 5 = 2^2 \times 5
\]

Prime Factorization Tree:
```
20
/ \
2 10
/ \
2 5
```

---

3. Number: 63

- Start with 63.
- Divide by the smallest prime number, which is 3:
\[
63 \div 3 = 21
\]
- Now, factorize 21:
\[
21 \div 3 = 7
\]
- Finally, 7 is a prime number.
- The prime factorization of 63 is:
\[
63 = 3 \times 3 \times 7 = 3^2 \times 7
\]

Prime Factorization Tree:
```
63
/ \
3 21
/ \
3 7
```

---

4. Number: 68

- Start with 68.
- Divide by the smallest prime number, which is 2:
\[
68 \div 2 = 34
\]
- Now, factorize 34:
\[
34 \div 2 = 17
\]
- Finally, 17 is a prime number.
- The prime factorization of 68 is:
\[
68 = 2 \times 2 \times 17 = 2^2 \times 17
\]

Prime Factorization Tree:
```
68
/ \
2 34
/ \
2 17
```

---

5. Number: 75

- Start with 75.
- Divide by the smallest prime number, which is 3:
\[
75 \div 3 = 25
\]
- Now, factorize 25:
\[
25 \div 5 = 5
\]
- Finally, 5 is a prime number.
- The prime factorization of 75 is:
\[
75 = 3 \times 5 \times 5 = 3 \times 5^2
\]

Prime Factorization Tree:
```
75
/ \
3 25
/ \
5 5
```

---

6. Number: 52

- Start with 52.
- Divide by the smallest prime number, which is 2:
\[
52 \div 2 = 26
\]
- Now, factorize 26:
\[
26 \div 2 = 13
\]
- Finally, 13 is a prime number.
- The prime factorization of 52 is:
\[
52 = 2 \times 2 \times 13 = 2^2 \times 13
\]

Prime Factorization Tree:
```
52
/ \
2 26
/ \
2 13
```

---

7. Number: 42

- Start with 42.
- Divide by the smallest prime number, which is 2:
\[
42 \div 2 = 21
\]
- Now, factorize 21:
\[
21 \div 3 = 7
\]
- Finally, 7 is a prime number.
- The prime factorization of 42 is:
\[
42 = 2 \times 3 \times 7
\]

Prime Factorization Tree:
```
42
/ \
2 21
/ \
3 7
```

---

8. Number: 45

- Start with 45.
- Divide by the smallest prime number, which is 3:
\[
45 \div 3 = 15
\]
- Now, factorize 15:
\[
15 \div 3 = 5
\]
- Finally, 5 is a prime number.
- The prime factorization of 45 is:
\[
45 = 3 \times 3 \times 5 = 3^2 \times 5
\]

Prime Factorization Tree:
```
45
/ \
3 15
/ \
3 5
```

---

9. Number: 99

- Start with 99.
- Divide by the smallest prime number, which is 3:
\[
99 \div 3 = 33
\]
- Now, factorize 33:
\[
33 \div 3 = 11
\]
- Finally, 11 is a prime number.
- The prime factorization of 99 is:
\[
99 = 3 \times 3 \times 11 = 3^2 \times 11
\]

Prime Factorization Tree:
```
99
/ \
3 33
/ \
3 11
```

---

Final Answers:

\[
\boxed{
\begin{aligned}
27 &= 3 \times 3 \times 3 = 3^3 \\
20 &= 2 \times 2 \times 5 = 2^2 \times 5 \\
63 &= 3 \times 3 \times 7 = 3^2 \times 7 \\
68 &= 2 \times 2 \times 17 = 2^2 \times 17 \\
75 &= 3 \times 5 \times 5 = 3 \times 5^2 \\
52 &= 2 \times 2 \times 13 = 2^2 \times 13 \\
42 &= 2 \times 3 \times 7 \\
45 &= 3 \times 3 \times 5 = 3^2 \times 5 \\
99 &= 3 \times 3 \times 11 = 3^2 \times 11
\end{aligned}
}
\]