Prime Factorization Worksheet for students to practice factoring numbers and identifying primes.

To solve the problem, we need to factor each given number into its prime factors and determine whether each number is prime. Let's go through each number step by step.

Prime Factorization and Primality Check

#### 1) 46
- Factorization:
- 46 is even, so divide by 2: \( 46 \div 2 = 23 \)
- 23 is a prime number.
- Prime factorization: \( 46 = 2 \times 23 \)
- Primality: Not prime (it has factors other than 1 and itself).

#### 2) 67
- Factorization:
- 67 is not divisible by 2, 3, 5, or any other prime number less than its square root (\(\sqrt{67} \approx 8.2\)).
- Prime factorization: \( 67 = 67 \) (it is a prime number).
- Primality: Prime.

#### 3) 9
- Factorization:
- 9 is not even, but it is divisible by 3: \( 9 \div 3 = 3 \)
- Prime factorization: \( 9 = 3 \times 3 = 3^2 \)
- Primality: Not prime.

#### 4) 2
- Factorization:
- 2 is a prime number.
- Prime factorization: \( 2 = 2 \)
- Primality: Prime.

#### 5) 4
- Factorization:
- 4 is even, so divide by 2: \( 4 \div 2 = 2 \)
- Prime factorization: \( 4 = 2 \times 2 = 2^2 \)
- Primality: Not prime.

#### 6) 80
- Factorization:
- 80 is even, so divide by 2: \( 80 \div 2 = 40 \)
- 40 is even, so divide by 2: \( 40 \div 2 = 20 \)
- 20 is even, so divide by 2: \( 20 \div 2 = 10 \)
- 10 is even, so divide by 2: \( 10 \div 2 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 80 = 2 \times 2 \times 2 \times 2 \times 5 = 2^4 \times 5 \)
- Primality: Not prime.

#### 7) 40
- Factorization:
- 40 is even, so divide by 2: \( 40 \div 2 = 20 \)
- 20 is even, so divide by 2: \( 20 \div 2 = 10 \)
- 10 is even, so divide by 2: \( 10 \div 2 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5 \)
- Primality: Not prime.

#### 8) 76
- Factorization:
- 76 is even, so divide by 2: \( 76 \div 2 = 38 \)
- 38 is even, so divide by 2: \( 38 \div 2 = 19 \)
- 19 is a prime number.
- Prime factorization: \( 76 = 2 \times 2 \times 19 = 2^2 \times 19 \)
- Primality: Not prime.

#### 9) 27
- Factorization:
- 27 is not even, but it is divisible by 3: \( 27 \div 3 = 9 \)
- 9 is divisible by 3: \( 9 \div 3 = 3 \)
- Prime factorization: \( 27 = 3 \times 3 \times 3 = 3^3 \)
- Primality: Not prime.

#### 10) 16
- Factorization:
- 16 is even, so divide by 2: \( 16 \div 2 = 8 \)
- 8 is even, so divide by 2: \( 8 \div 2 = 4 \)
- 4 is even, so divide by 2: \( 4 \div 2 = 2 \)
- 2 is a prime number.
- Prime factorization: \( 16 = 2 \times 2 \times 2 \times 2 = 2^4 \)
- Primality: Not prime.

#### 11) 25
- Factorization:
- 25 is not even, but it is divisible by 5: \( 25 \div 5 = 5 \)
- Prime factorization: \( 25 = 5 \times 5 = 5^2 \)
- Primality: Not prime.

#### 12) 21
- Factorization:
- 21 is not even, but it is divisible by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Prime factorization: \( 21 = 3 \times 7 \)
- Primality: Not prime.

#### 13) 18
- Factorization:
- 18 is even, so divide by 2: \( 18 \div 2 = 9 \)
- 9 is divisible by 3: \( 9 \div 3 = 3 \)
- Prime factorization: \( 18 = 2 \times 3 \times 3 = 2 \times 3^2 \)
- Primality: Not prime.

#### 14) 22
- Factorization:
- 22 is even, so divide by 2: \( 22 \div 2 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 22 = 2 \times 11 \)
- Primality: Not prime.

#### 15) 30
- Factorization:
- 30 is even, so divide by 2: \( 30 \div 2 = 15 \)
- 15 is divisible by 3: \( 15 \div 3 = 5 \)
- 5 is a prime number.
- Prime factorization: \( 30 = 2 \times 3 \times 5 \)
- Primality: Not prime.

#### 16) 39
- Factorization:
- 39 is not even, but it is divisible by 3: \( 39 \div 3 = 13 \)
- 13 is a prime number.
- Prime factorization: \( 39 = 3 \times 13 \)
- Primality: Not prime.

#### 17) 55
- Factorization:
- 55 is not even, but it is divisible by 5: \( 55 \div 5 = 11 \)
- 11 is a prime number.
- Prime factorization: \( 55 = 5 \times 11 \)
- Primality: Not prime.

#### 18) 53
- Factorization:
- 53 is not divisible by 2, 3, 5, or any other prime number less than its square root (\(\sqrt{53} \approx 7.3\)).
- Prime factorization: \( 53 = 53 \) (it is a prime number).
- Primality: Prime.

#### 19) 47
- Factorization:
- 47 is not divisible by 2, 3, 5, or any other prime number less than its square root (\(\sqrt{47} \approx 6.9\)).
- Prime factorization: \( 47 = 47 \) (it is a prime number).
- Primality: Prime.

#### 20) 32
- Factorization:
- 32 is even, so divide by 2: \( 32 \div 2 = 16 \)
- 16 is even, so divide by 2: \( 16 \div 2 = 8 \)
- 8 is even, so divide by 2: \( 8 \div 2 = 4 \)
- 4 is even, so divide by 2: \( 4 \div 2 = 2 \)
- 2 is a prime number.
- Prime factorization: \( 32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5 \)
- Primality: Not prime.

Final Answers

1. \( 46 = 2 \times 23 \) (Not prime)
2. \( 67 = 67 \) (Prime)
3. \( 9 = 3^2 \) (Not prime)
4. \( 2 = 2 \) (Prime)
5. \( 4 = 2^2 \) (Not prime)
6. \( 80 = 2^4 \times 5 \) (Not prime)
7. \( 40 = 2^3 \times 5 \) (Not prime)
8. \( 76 = 2^2 \times 19 \) (Not prime)
9. \( 27 = 3^3 \) (Not prime)
10. \( 16 = 2^4 \) (Not prime)
11. \( 25 = 5^2 \) (Not prime)
12. \( 21 = 3 \times 7 \) (Not prime)
13. \( 18 = 2 \times 3^2 \) (Not prime)
14. \( 22 = 2 \times 11 \) (Not prime)
15. \( 30 = 2 \times 3 \times 5 \) (Not prime)
16. \( 39 = 3 \times 13 \) (Not prime)
17. \( 55 = 5 \times 11 \) (Not prime)
18. \( 53 = 53 \) (Prime)
19. \( 47 = 47 \) (Prime)
20. \( 32 = 2^5 \) (Not prime)

Boxed Answer

\[
\boxed{
\begin{array}{ll}
1) & 46 = 2 \times 23 \quad \text{(Not prime)} \\
2) & 67 = 67 \quad \text{(Prime)} \\
3) & 9 = 3^2 \quad \text{(Not prime)} \\
4) & 2 = 2 \quad \text{(Prime)} \\
5) & 4 = 2^2 \quad \text{(Not prime)} \\
6) & 80 = 2^4 \times 5 \quad \text{(Not prime)} \\
7) & 40 = 2^3 \times 5 \quad \text{(Not prime)} \\
8) & 76 = 2^2 \times 19 \quad \text{(Not prime)} \\
9) & 27 = 3^3 \quad \text{(Not prime)} \\
10) & 16 = 2^4 \quad \text{(Not prime)} \\
11) & 25 = 5^2 \quad \text{(Not prime)} \\
12) & 21 = 3 \times 7 \quad \text{(Not prime)} \\
13) & 18 = 2 \times 3^2 \quad \text{(Not prime)} \\
14) & 22 = 2 \times 11 \quad \text{(Not prime)} \\
15) & 30 = 2 \times 3 \times 5 \quad \text{(Not prime)} \\
16) & 39 = 3 \times 13 \quad \text{(Not prime)} \\
17) & 55 = 5 \times 11 \quad \text{(Not prime)} \\
18) & 53 = 53 \quad \text{(Prime)} \\
19) & 47 = 47 \quad \text{(Prime)} \\
20) & 32 = 2^5 \quad \text{(Not prime)} \\
\end{array}
}
\]