Sum of two different prime numbers is a ________.
Hint 1: Consider the parity of prime numbers: 2 is the only even prime, all others are odd.
Hint 2: Try adding 2 to another prime (e.g., , ).
Hint 3: Try adding two odd primes (e.g., ). Note that the sum of two odd primes is always even (hence composite).
Step 1 (Analyze Odd/Even Parity): All prime numbers except are odd. When we add two different prime numbers, we have two possible cases:
Case 1: One of the primes is 2.
Adding 2 to any odd prime results in an odd number .
If , the sum is (which is prime).
If , the sum is (which is composite).
Case 2: Both primes are odd.
Adding two odd prime numbers always results in an even number (since odd + odd = even). Since both primes are different and odd, their sum is at least , which is an even number greater than 2, and therefore always composite.
Step 2 (Conclusion): Since the sum can be prime (like 5) or composite (like 9 or 8), the sum of two different prime numbers is a prime or composite number.
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