Can a Number be both Prime and Composite?

Answer: No, a number cannot be both prime and composite because prime numbers have exactly two distinct positive divisors, while composite numbers have more than two distinct positive divisors.

Let’s break it down:

1. Prime Numbers:
   • Prime numbers are natural numbers greater than 1.
   • They have precisely two distinct positive divisors: 1 and the number itself.
   • Examples include 2, 3, 5, 7, 11, and so on.
2. Composite Numbers:
   • Composite numbers are natural numbers greater than 1.
   • They have more than two distinct positive divisors, including 1 and the number itself.
   • Examples include 4 (divisible by 1, 2, and 4), 6 (divisible by 1, 2, 3, and 6), and 8 (divisible by 1, 2, 4, and 8).
3. Mutual Exclusivity:
   • The definitions of prime and composite numbers are mutually exclusive.
   • A number cannot meet both criteria simultaneously because the number of distinct positive divisors is a fundamental property that distinguishes primes from composites.
4. Distinct Categories:
   • Prime numbers and composite numbers represent distinct categories in number theory.
   • A number is either prime or composite, but not both.

In summary, a number cannot be both prime and composite due to the mutually exclusive definitions and fundamental properties associated with these categories in number theory.