Prime and Composite Numbers Worksheets - 15 Worksheets Library - Free Printable

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

Definitions:

- Prime Number: A number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
- Composite Number: A number greater than 1 that has more than two distinct positive divisors.

Step-by-Step Solution:

#### 1. 25
- Factors: \(1, 5, 25\)
- Since 25 has more than two factors (1, 5, 25), it is composite.
- Result: Composite

#### 2. 36
- Factors: To find the factors of 36, we check divisibility:
- \(36 \div 1 = 36\)
- \(36 \div 2 = 18\)
- \(36 \div 3 = 12\)
- \(36 \div 4 = 9\)
- \(36 \div 6 = 6\)
- Stop here since the factors start repeating.
- Factors: \(1, 2, 3, 4, 6, 9, 12, 18, 36\)
- Since 36 has more than two factors, it is composite.
- Result: Composite

#### 3. 43
- Factors: To check if 43 is prime, we test divisibility by prime numbers less than \(\sqrt{43} \approx 6.5\):
- Not divisible by 2 (it's odd).
- Not divisible by 3 (sum of digits \(4 + 3 = 7\) is not divisible by 3).
- Not divisible by 5 (does not end in 0 or 5).
- Not divisible by any other primes less than \(\sqrt{43}\).
- Factors: \(1, 43\)
- Since 43 has exactly two factors, it is prime.
- Result: Prime

#### 4. 47
- Factors: To check if 47 is prime, we test divisibility by prime numbers less than \(\sqrt{47} \approx 6.8\):
- Not divisible by 2 (it's odd).
- Not divisible by 3 (sum of digits \(4 + 7 = 11\) is not divisible by 3).
- Not divisible by 5 (does not end in 0 or 5).
- Not divisible by any other primes less than \(\sqrt{47}\).
- Factors: \(1, 47\)
- Since 47 has exactly two factors, it is prime.
- Result: Prime

#### 5. 51
- Factors: To find the factors of 51, we check divisibility:
- \(51 \div 1 = 51\)
- \(51 \div 3 = 17\)
- \(51 \div 17 = 3\)
- Stop here since the factors start repeating.
- Factors: \(1, 3, 17, 51\)
- Since 51 has more than two factors, it is composite.
- Result: Composite

#### 6. 55
- Factors: To find the factors of 55, we check divisibility:
- \(55 \div 1 = 55\)
- \(55 \div 5 = 11\)
- \(55 \div 11 = 5\)
- Stop here since the factors start repeating.
- Factors: \(1, 5, 11, 55\)
- Since 55 has more than two factors, it is composite.
- Result: Composite

#### 7. 59
- Factors: To check if 59 is prime, we test divisibility by prime numbers less than \(\sqrt{59} \approx 7.7\):
- Not divisible by 2 (it's odd).
- Not divisible by 3 (sum of digits \(5 + 9 = 14\) is not divisible by 3).
- Not divisible by 5 (does not end in 0 or 5).
- Not divisible by 7 (\(59 \div 7 \approx 8.43\)).
- Factors: \(1, 59\)
- Since 59 has exactly two factors, it is prime.
- Result: Prime

#### 8. 60
- Factors: To find the factors of 60, we check divisibility:
- \(60 \div 1 = 60\)
- \(60 \div 2 = 30\)
- \(60 \div 3 = 20\)
- \(60 \div 4 = 15\)
- \(60 \div 5 = 12\)
- \(60 \div 6 = 10\)
- Stop here since the factors start repeating.
- Factors: \(1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\)
- Since 60 has more than two factors, it is composite.
- Result: Composite

#### 9. 61
- Factors: To check if 61 is prime, we test divisibility by prime numbers less than \(\sqrt{61} \approx 7.8\):
- Not divisible by 2 (it's odd).
- Not divisible by 3 (sum of digits \(6 + 1 = 7\) is not divisible by 3).
- Not divisible by 5 (does not end in 0 or 5).
- Not divisible by 7 (\(61 \div 7 \approx 8.71\)).
- Factors: \(1, 61\)
- Since 61 has exactly two factors, it is prime.
- Result: Prime

Final Table:

| Number | Factors | Prime or Composite |
|--------|----------------------------------|--------------------|
| 25 | 1, 5, 25 | Composite |
| 36 | 1, 2, 3, 4, 6, 9, 12, 18, 36 | Composite |
| 43 | 1, 43 | Prime |
| 47 | 1, 47 | Prime |
| 51 | 1, 3, 17, 51 | Composite |
| 55 | 1, 5, 11, 55 | Composite |
| 59 | 1, 59 | Prime |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 | Composite |
| 61 | 1, 61 | Prime |

Final Answer:

\[
\boxed{
\begin{array}{|c|c|c|}
\hline
\text{Number} & \text{Factors} & \text{Prime or Composite} \\
\hline
25 & 1, 5, 25 & \text{Composite} \\
36 & 1, 2, 3, 4, 6, 9, 12, 18, 36 & \text{Composite} \\
43 & 1, 43 & \text{Prime} \\
47 & 1, 47 & \text{Prime} \\
51 & 1, 3, 17, 51 & \text{Composite} \\
55 & 1, 5, 11, 55 & \text{Composite} \\
59 & 1, 59 & \text{Prime} \\
60 & 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 & \text{Composite} \\
61 & 1, 61 & \text{Prime} \\
\hline
\end{array}
}
\]