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:
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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.
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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).
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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.
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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.