Yes, composite numbers always have prime divisors.
A composite number is a positive integer that has more than two distinct positive divisors: 1, itself, and at least one other factor. In other words, it's a number that can be formed by multiplying two smaller positive integers. Prime numbers, on the other hand, only have two divisors: 1 and themselves.
Here's why composite numbers must have prime divisors:
- Definition of Composite: By definition, a composite number n can be written as n = a b, where a and b are integers greater than 1.
- Factorization: The factors a and b are either prime numbers themselves, or they are composite and can be further factored.
- Prime Factorization Theorem: The fundamental theorem of arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.
- Conclusion: Therefore, since a composite number can be factored into smaller numbers, and those smaller numbers are either prime or can be further factored into primes, a composite number must have prime numbers as divisors.
Examples:
- The number 12 is composite because it can be written as 2 x 6 or 3 x 4. The prime factors of 12 are 2 and 3 (12 = 2 x 2 x 3 = 22 x 3).
- The number 20 is composite because it can be written as 4 x 5. The prime factors of 20 are 2 and 5 (20 = 2 x 2 x 5 = 22 x 5).
- The number 35 is composite because it can be written as 5 x 7. The prime factors of 35 are 5 and 7.
In essence, if a number is composite, it must be divisible by at least one prime number. If it weren't, it would be a prime number itself.
