Prime Numbers Worksheet 3 - Fill in missing numbers to complete prime factorization exercises.

Problem Description:

The task involves completing the prime factorization of given numbers by filling in the missing numbers in the provided tables. Prime factorization is the process of expressing a number as a product of its prime factors.

Solution Approach:

1. Understand Prime Factorization:
- A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
- Prime factorization breaks down a number into its prime factors, which are multiplied together to give the original number.

2. Steps for Prime Factorization:
- Start with the smallest prime number (2) and divide the given number by it if possible.
- Continue dividing by prime numbers until the quotient is 1.
- Write the prime factors in the table as shown in the example.

3. Solve Each Problem:
- Follow the steps for each number provided in the worksheet.

---

Detailed Solutions:

#### Problem 1: 72
```
2 | 72
2 | 36
2 | 18
3 | 9
3 | 3
| 1
```
- Explanation:
- Divide 72 by 2: \( 72 \div 2 = 36 \)
- Divide 36 by 2: \( 36 \div 2 = 18 \)
- Divide 18 by 2: \( 18 \div 2 = 9 \)
- Divide 9 by 3: \( 9 \div 3 = 3 \)
- Divide 3 by 3: \( 3 \div 3 = 1 \)

- Prime Factorization: \( 72 = 2 \times 2 \times 2 \times 3 \times 3 \)

#### Problem 2: 120
```
2 | 120
2 | 60
2 | 30
3 | 15
5 | 5
| 1
```
- Explanation:
- Divide 120 by 2: \( 120 \div 2 = 60 \)
- Divide 60 by 2: \( 60 \div 2 = 30 \)
- Divide 30 by 2: \( 30 \div 2 = 15 \)
- Divide 15 by 3: \( 15 \div 3 = 5 \)
- Divide 5 by 5: \( 5 \div 5 = 1 \)

- Prime Factorization: \( 120 = 2 \times 2 \times 2 \times 3 \times 5 \)

#### Problem 3: 252
```
2 | 252
2 | 126
3 | 63
3 | 21
7 | 7
| 1
```
- Explanation:
- Divide 252 by 2: \( 252 \div 2 = 126 \)
- Divide 126 by 2: \( 126 \div 2 = 63 \)
- Divide 63 by 3: \( 63 \div 3 = 21 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- Divide 7 by 7: \( 7 \div 7 = 1 \)

- Prime Factorization: \( 252 = 2 \times 2 \times 3 \times 3 \times 7 \)

#### Problem 4: 168
```
2 | 168
2 | 84
2 | 42
3 | 21
7 | 7
| 1
```
- Explanation:
- Divide 168 by 2: \( 168 \div 2 = 84 \)
- Divide 84 by 2: \( 84 \div 2 = 42 \)
- Divide 42 by 2: \( 42 \div 2 = 21 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- Divide 7 by 7: \( 7 \div 7 = 1 \)

- Prime Factorization: \( 168 = 2 \times 2 \times 2 \times 3 \times 7 \)

#### Problem 5: 420
```
2 | 420
2 | 210
3 | 105
5 | 35
7 | 7
| 1
```
- Explanation:
- Divide 420 by 2: \( 420 \div 2 = 210 \)
- Divide 210 by 2: \( 210 \div 2 = 105 \)
- Divide 105 by 3: \( 105 \div 3 = 35 \)
- Divide 35 by 5: \( 35 \div 5 = 7 \)
- Divide 7 by 7: \( 7 \div 7 = 1 \)

- Prime Factorization: \( 420 = 2 \times 2 \times 3 \times 5 \times 7 \)

#### Problem 6: 210
```
2 | 210
3 | 105
5 | 35
7 | 7
| 1
```
- Explanation:
- Divide 210 by 2: \( 210 \div 2 = 105 \)
- Divide 105 by 3: \( 105 \div 3 = 35 \)
- Divide 35 by 5: \( 35 \div 5 = 7 \)
- Divide 7 by 7: \( 7 \div 7 = 1 \)

- Prime Factorization: \( 210 = 2 \times 3 \times 5 \times 7 \)

#### Problem 7: 378
```
2 | 378
3 | 189
3 | 63
3 | 21
7 | 7
| 1
```
- Explanation:
- Divide 378 by 2: \( 378 \div 2 = 189 \)
- Divide 189 by 3: \( 189 \div 3 = 63 \)
- Divide 63 by 3: \( 63 \div 3 = 21 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- Divide 7 by 7: \( 7 \div 7 = 1 \)

- Prime Factorization: \( 378 = 2 \times 3 \times 3 \times 3 \times 7 \)

#### Problem 8: 2310
```
2 | 2310
3 | 1155
5 | 231
7 | 33
11 | 11
11 | 1
```
- Explanation:
- Divide 2310 by 2: \( 2310 \div 2 = 1155 \)
- Divide 1155 by 3: \( 1155 \div 3 = 385 \)
- Divide 385 by 5: \( 385 \div 5 = 77 \)
- Divide 77 by 7: \( 77 \div 7 = 11 \)
- Divide 11 by 11: \( 11 \div 11 = 1 \)

- Prime Factorization: \( 2310 = 2 \times 3 \times 5 \times 7 \times 11 \)

---

Final Answers:

1. \( 72 = 2 \times 2 \times 2 \times 3 \times 3 \)
2. \( 120 = 2 \times 2 \times 2 \times 3 \times 5 \)
3. \( 252 = 2 \times 2 \times 3 \times 3 \times 7 \)
4. \( 168 = 2 \times 2 \times 2 \times 3 \times 7 \)
5. \( 420 = 2 \times 2 \times 3 \times 5 \times 7 \)
6. \( 210 = 2 \times 3 \times 5 \times 7 \)
7. \( 378 = 2 \times 3 \times 3 \times 3 \times 7 \)
8. \( 2310 = 2 \times 3 \times 5 \times 7 \times 11 \)

\boxed{
\begin{aligned}
1. & \ 2 \times 2 \times 2 \times 3 \times 3 \\
2. & \ 2 \times 2 \times 2 \times 3 \times 5 \\
3. & \ 2 \times 2 \times 3 \times 3 \times 7 \\
4. & \ 2 \times 2 \times 2 \times 3 \times 7 \\
5. & \ 2 \times 2 \times 3 \times 5 \times 7 \\
6. & \ 2 \times 3 \times 5 \times 7 \\
7. & \ 2 \times 3 \times 3 \times 3 \times 7 \\
8. & \ 2 \times 3 \times 5 \times 7 \times 11
\end{aligned}
}