The prime numbers are the natural whole numbers (1, 2, 3, 4, 5, 6, etc.) which are only divisible by 1 and themselves, which leads to the elimination in the previous list of the composite numbers. Composite numbers are products of two numbers like 4 (2 x 2), 6 (2 x 3), etc. So – is 1 a prime number?
With this definition, 1 would be prime. It is eliminated, however, by adding that a prime number must have two distinct divisors: 1 and itself. The reason is deeper than it can seem. Let’s see why.
If we exclude 1 from the set of prime numbers, we can prove a theorem: Any natural integer greater than 2 is the product of a finite number of prime numbers and uniquely, in the order of factors.
So, 530 = 2 x 5 x 53
To prove this theorem, the essential point is to show that every number is either prime or divisible by a prime number… which is obvious. Applying this remark iteratively, we arrive at our theorem.
If we do not exclude 1, any decomposition is multiple since we can add as many factors 1 as we want without changing the result.