Complete the prime factor tree for each numeral.

To complete the prime factor trees for each numeral, we need to break down each number into its prime factors step by step. Let's solve each problem systematically.

---

1) Prime Factor Tree for 45

The given tree is:
```
45
/ \
? 3
/ \
3 ?
```

#### Steps:
1. Start with 45.
2. Divide 45 by the smallest prime number, which is 3:
\( 45 \div 3 = 15 \). So, the first split is \( 45 = 3 \times 15 \).
3. Now, factorize 15. Divide 15 by 3:
\( 15 \div 3 = 5 \). So, \( 15 = 3 \times 5 \).
4. The number 5 is already a prime number.

#### Final Tree:
```
45
/ \
15 3
/ \
3 5
```

---

2) Prime Factor Tree for 24

The given tree is:
```
24
/ \
4 ?
/ \
2 ?
\
2
```

#### Steps:
1. Start with 24.
2. Divide 24 by the smallest prime number, which is 2:
\( 24 \div 2 = 12 \). So, the first split is \( 24 = 2 \times 12 \).
3. Now, factorize 12. Divide 12 by 2:
\( 12 \div 2 = 6 \). So, \( 12 = 2 \times 6 \).
4. Next, factorize 6. Divide 6 by 2:
\( 6 \div 2 = 3 \). So, \( 6 = 2 \times 3 \).
5. The number 3 is already a prime number.

#### Final Tree:
```
24
/ \
4 6
/ \ / \
2 2 2 3
```

---

3) Prime Factor Tree for 32

The given tree is:
```
32
/ \
? 4
/ \
4 2
/ \
2 2
```

#### Steps:
1. Start with 32.
2. Divide 32 by the smallest prime number, which is 2:
\( 32 \div 2 = 16 \). So, the first split is \( 32 = 2 \times 16 \).
3. Now, factorize 16. Divide 16 by 2:
\( 16 \div 2 = 8 \). So, \( 16 = 2 \times 8 \).
4. Next, factorize 8. Divide 8 by 2:
\( 8 \div 2 = 4 \). So, \( 8 = 2 \times 4 \).
5. Finally, factorize 4. Divide 4 by 2:
\( 4 \div 2 = 2 \). So, \( 4 = 2 \times 2 \).

#### Final Tree:
```
32
/ \
2 16
/ \
2 8
/ \
2 4
/ \
2 2
```

---

4) Prime Factor Tree for 12

The given tree is:
```
12
/ \
? 6
/ \
2 ?
```

#### Steps:
1. Start with 12.
2. Divide 12 by the smallest prime number, which is 2:
\( 12 \div 2 = 6 \). So, the first split is \( 12 = 2 \times 6 \).
3. Now, factorize 6. Divide 6 by 2:
\( 6 \div 2 = 3 \). So, \( 6 = 2 \times 3 \).
4. The number 3 is already a prime number.

#### Final Tree:
```
12
/ \
2 6
/ \
2 3
```

---

5) Prime Factor Tree for 8

The given tree is:
```
8
/ \
2 ?
/ \
? 2
```

#### Steps:
1. Start with 8.
2. Divide 8 by the smallest prime number, which is 2:
\( 8 \div 2 = 4 \). So, the first split is \( 8 = 2 \times 4 \).
3. Now, factorize 4. Divide 4 by 2:
\( 4 \div 2 = 2 \). So, \( 4 = 2 \times 2 \).

#### Final Tree:
```
8
/ \
2 4
/ \
2 2
```

---

6) Prime Factor Tree for 42

The given tree is:
```
42
/ \
? 14
\
7
\
?
```

#### Steps:
1. Start with 42.
2. Divide 42 by the smallest prime number, which is 2:
\( 42 \div 2 = 21 \). So, the first split is \( 42 = 2 \times 21 \).
3. Now, factorize 21. Divide 21 by 3:
\( 21 \div 3 = 7 \). So, \( 21 = 3 \times 7 \).
4. The numbers 3 and 7 are already prime numbers.

#### Final Tree:
```
42
/ \
2 21
\
3
\
7
```

---

Final Answers:

1.
```
45
/ \
15 3
/ \
3 5
```
2.
```
24
/ \
4 6
/ \ / \
2 2 2 3
```
3.
```
32
/ \
2 16
/ \
2 8
/ \
2 4
/ \
2 2
```
4.
```
12
/ \
2 6
/ \
2 3
```
5.
```
8
/ \
2 4
/ \
2 2
```
6.
```
42
/ \
2 21
\
3
\
7
```

Boxed Final Answer:

\[
\boxed{
\begin{array}{l}
1) \begin{array}{c}
45 \\
/ \backslash \\
15 3 \\
/ \backslash \\
3 5
\end{array} \\
2) \begin{array}{c}
24 \\
/ \backslash \\
4 6 \\
/ \backslash / \backslash \\
2 2 2 3
\end{array} \\
3) \begin{array}{c}
32 \\
/ \backslash \\
2 16 \\
/ \backslash \\
2 8 \\
/ \backslash \\
2 4 \\
/ \backslash \\
2 2
\end{array} \\
4) \begin{array}{c}
12 \\
/ \backslash \\
2 6 \\
/ \backslash \\
2 3
\end{array} \\
5) \begin{array}{c}
8 \\
/ \backslash \\
2 4 \\
/ \backslash \\
2 2
\end{array} \\
6) \begin{array}{c}
42 \\
/ \backslash \\
2 21 \\
\backslash \\
3 \\
\backslash \\
7
\end{array}
\end{array}
}
\]