Let's break down the problem logically:

1. Sum of $p$ even numbers is even:

   • When you add any number of even numbers, the result is always even. This is because the sum of any two even numbers is still even. So the sum of $p$ even numbers will always be even.

2. Sum of $q$ odd numbers is odd:

   • The sum of odd numbers depends on how many odd numbers are being added. Specifically:
     • The sum of an even number of odd numbers is even.
     • The sum of an odd number of odd numbers is odd.
   • Since the sum of $q$ odd numbers is given as odd, this implies that $q$ must be odd (since only the sum of an odd number of odd numbers is odd).

Now let's examine each option:

• Option (a): $p + q$ is even:

  • We don't know whether $p$ is even or odd, but we know $q$ is odd. So $p + q$ could be either even or odd, depending on whether $p$ is even or odd.
  • Hence, this statement is not necessarily true.

• Option (b): $p + q$ is odd:

  • Again, we don't know whether $p$ is even or odd. So $p + q$ could be either even or odd.
  • Hence, this statement is not necessarily true.

• Option (c): $pq$ is even:

  • We know $q$ is odd, but we don't know whether $p$ is even or odd.
  • If $p$ is even, then $pq$ will definitely be even (because even times any number is even).
  • If $p$ is odd, then $pq$ will be odd (because odd times odd is odd).
  • So, $pq$ is not necessarily even.

• Option (d): None:

  • Since none of the previous options must necessarily be true, the correct answer is (d) None.

Would you like more details on any part of this solution?

Here are 5 related questions:

1. What is the sum of two odd numbers?
2. How does multiplying even and odd numbers affect the result?
3. Can the sum of both even and odd numbers be odd?
4. What is the result of multiplying an even and odd number?
5. Why does the sum of an even number of odd numbers become even?

Tip: When working with even and odd numbers, keep in mind that even numbers are multiples of 2, while odd numbers cannot be divided evenly by 2. This can simplify problem-solving.