Why are some numbers not prime nor composite?

I just am wondering why? I never got taught this. Is it because You can't multiply them by anything?Why are some numbers not prime nor composite?
It depends on the definition of prime and composite.





If you define composite as not prime, then all numbers will be one or the other.





If you define composite as having at least two prime factors, then there are numbers like 1 and 0 which are neither prime nor composite. (Some people will incorrectly say that 1 is prime. It's not true.)Why are some numbers not prime nor composite?
It's all about definition. Zero and One are unique numbers in that they can be proven mathematically not to fall into either category.





Prime: a number that only has two integer factors, 1 and itself. One has only one as its factor, so it doesn't meet the definition. Zero can be multiplied by any integer and still be zero, so it has an infinite number of integer factors.





Composite: These numbers have many definitions (all saying the same thing a different way), but I like the one that says that a composite number has to have at least one other integer factor besides itself.


- this definition excludes one because itself (1) is its only factor, and


- it excludes zero because zero is always a factor of itself (0x5, 0x100, etc.)





12 is composite because, in addition to having factors 12 x 1, it also has other factors, like 6 x 2, that do not include the number 12.
The definition of prime is that an integer has two and only two unique factors: 1 and itself. For example, 2 is prime since the only factors of 2 are 1 and 2. 4 is not prime since the factors of 4 are 1, 2, and 4.





The definition of a composite number is a number than can be written as a product of two integers such that neither factor is 1 or itself. From the example above, we can easily see that 4 is composite since 2 * 2 = 4.





So what about 0 and 1? 0 would appear to be prime since 0 * 1 = 0. But 0 is not prime since there exists no expression a * b = 0, where a and b are non zero. Also, the solution to a * 0 = 0 is all real numbers. That is, 0 has an infinite number of factors. This violates the definition of a prime since other factors are there and violates the definition of a composite number since there is no combination that involves factors other than 0.





1 cannot be prime since the only equation that equals 1 is 1 * 1 = 1. This violates the definition of a prime number since the factors must be UNIQUE. This also violates the definition of a composite number since one of the factors must not be 1.





Considering this another way, take the prime factorization of any composite number (say, 12 = 2 * 2 * 3). If 1 were a prime, we could also rewrite such factorizations as:





12 = 2 * 2 * 3


12 = 1 * 2 * 2 * 3


12 = 1 * 1 * 2 * 2 * 3


...





And so on and so forth until we get:





12 = 1^infinity * 2 * 2 * 3





Again, this violates the uniqueness requirement on the definition of a prime. Therefore, 1 is not prime.





Hope this discussion was enlightening and interesting.