# Q. Consider the following in respect of prime number p and composite number c.

1. p+c / p-c can be even.

2. 2p+ c can be odd.

3. pc can be odd.

Which of the statements given above are correct?

(a) 1 and 2 only

(b) 2 and 3 only

(c) 1 and 3 only

(d) 1, 2 and 3

Correct Answer: (d) 1, 2 and 3

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## Prime and Composite Number Analysis

To determine which statements are correct, let’s analyze each one individually:

### 1. **p + c / p – c can be even:**

- For this expression to be even, the numerator (p + c) and the denominator (p – c) must both be either even or odd.
- Prime numbers (p) are mostly odd, except for 2.
- Composite numbers (c) can be either even or odd.
- If p is odd and c is even, then:
- p + c is odd + even = odd
- p – c is odd – even = odd
- An odd number divided by an odd number can be even if the result is an integer.

- If p is 2 (even) and c is odd, then:
- p + c is even + odd = odd
- p – c is even – odd = odd
- An odd number divided by an odd number can be even if the result is an integer.

Therefore, it is possible for p + c / p – c to be even.

### 2. **2p + c can be odd:**

For 2p + c to be odd:

- 2p is always even because 2 times any integer is even.
- For the sum to be odd, c must be odd (even + odd = odd).
- Since composite numbers can be odd, this statement is true.

### 3. **pc can be odd:**

For the product pc to be odd:

- Both p and c must be odd.
- Since prime numbers (except 2) are mostly odd and composite numbers can be odd, this statement is true.

Given the analysis, all three statements are correct. Therefore, the correct answer is:

**(d) 1, 2 and 3**

**Prime Numbers:**

- A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
- In other words, a prime number is a number that can only be divided evenly (without a remainder) by 1 and itself.
- Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, 19, etc.
- Note that 2 is the only even prime number; all other prime numbers are odd.

### Composite Numbers:

- A composite number is a natural number greater than 1 that is not prime, meaning it has more than two positive divisors.
- In other words, a composite number can be divided evenly by numbers other than 1 and itself.
- Examples of composite numbers include 4, 6, 8, 9, 10, 12, 14, 15, etc.
- Composite numbers can be even or odd. For instance, 4 (even) and 9 (odd) are both composite numbers.