The numbers 1 and the number itself are not included in the prime factorization of a composite number.
Prime factorization is the process of breaking down a composite number into its prime factors. A composite number is a positive integer that has more than two distinct factors: 1, itself, and at least one other factor. The goal of prime factorization is to express a composite number as a product of its prime factors only.
Here's why 1 and the number itself are excluded:
-
1 is not a prime number: By definition, a prime number must have exactly two distinct positive divisors: 1 and itself. The number 1 only has one divisor (itself), so it does not fit the definition of a prime number. Since prime factorization deals exclusively with prime factors, 1 is not included.
-
The composite number itself: The purpose of prime factorization is to break the number down into its prime components. Including the number itself would be redundant and defeat the purpose of finding its prime factors.
Example:
Consider the composite number 12.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Prime factors of 12: 2, 3
- Prime factorization of 12: 2 x 2 x 3 (or 22 x 3)
As you can see, 1 and 12 are factors of 12, but they are not included in the prime factorization, which only consists of the prime factors 2 and 3.
In summary: Prime factorization aims to express a composite number as a product of its prime numbers only. Because 1 is not a prime number and including the number itself is redundant, they are excluded from the prime factorization.
