Question

Find the counter-example of the statement "Every natural number is either prime or composite''.

Answer 1

Counter-example of the given statement

Reasons why $1$ is not a prime:

1. A prime number is a positive integer that has no factors except $1$ and itself.
2. It is also a requirement that $1$ and “itself” must not be same. But in the case of $1$, it is same and thus $1$ is not a prime.
3. The fundamental theorem of arithmetic states that every integer can be uniquely written as a product of prime factors.
4. Including $1$ as a prime breaks this theorem as there will be infinte number of ways to represent a number. Example: $2=2\times 1=2\times 1\times 1=2\times 1^{84654}$

Thus, $1$ is not a prime number.

Reason why $1$ is not a composite:

1. A composite number is a positive integer that is divisible by other positive integers in addition to $1$ and itself.
2. But $1$ is not divisible by any other positive integer.

Thus, $1$ is not a composite number.

Therefore, the statement “The number $1$ is neither a prime nor a composite” is a counter-example of the given statement.