Consider the following statements in respect of two natural numbers p and q such that p is a prime number and q is a composite number:

1. p ×q can be an odd number.

2. q/p can be a prime number.

3. p + q can be a prime number.

Which of the above statements are correct?

The correct option is 1, 2 and 3.

Let us analyze each statement step-by-step by understanding the properties of prime and composite numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself, while a composite number is a natural number greater than 1 that has at least one divisor other than 1 and itself.

Statement 1: $p\times q$ can be an odd number.
For the product of two natural numbers to be odd, both numbers must be odd. Therefore, we need to check if we can select an odd prime number $p$ and an odd composite number $q$.
Let us choose:
- $p=3$ (which is an odd prime)
- $q=9$ (which is an odd composite)
Multiplying these two numbers yields:
$p\times q=3\times 9=27$
Since 27 is an odd number, Statement 1 is correct.

Statement 2: $\frac{q}{p}$ can be a prime number.
We need to find a composite number $q$ and a prime number $p$ such that their quotient is prime.
Let us choose:
- $p=3$ (which is a prime number)
- $q=9$ (which is a composite number)
Dividing them yields:
$\frac{q}{p}=\frac{9}{3}=3$
Since 3 is a prime number, Statement 2 is correct.

Statement 3: $p+q$ can be a prime number.
We need to see if the sum of a prime number $p$ and a composite number $q$ can result in a prime number.
Let us choose:
- $p=3$ (which is a prime number)
- $q=4$ (which is a composite number)
Adding them yields:
$p+q=3+4=7$
Since 7 is a prime number, Statement 3 is correct.

Because all three statements are correct, the correct option is indeed 1, 2 and 3.