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?

1. A. 1 and 2 only
2. B. 2 and 3 only
3. C. 1 and 3 only
4. D. 1, 2 and 3

Answer: D

Explanation

Let’s evaluate each statement:
1. (p+c)/(p-c) can be even.
Consider p = 11 (an odd prime) and c = 9 (an odd composite number).
p+c = 11+9 = 20 (even)
p-c = 11-9 = 2 (even)
(p+c)/(p-c) = 20/2 = 10, which is an even number. So, statement 1 is correct.
2. 2p+c can be odd.
Since p is an integer, 2p will always be an even number. For 2p+c to be odd, c must be an odd number (Even + Odd = Odd).
Consider p = 3 (an odd prime) and c = 9 (an odd composite number).
2p+c = 2(3) + 9 = 6 + 9 = 15, which is an odd number. So, statement 2 is correct.
3. pc can be odd.
For the product of two integers to be odd, both integers must be odd.
Consider p = 3 (an odd prime) and c = 9 (an odd composite number).
pc = 3 × 9 = 27, which is an odd number. So, statement 3 is correct.
Since all three statements can be true, the correct option is D.