Do prime numbers have prime factors?

Question

(This is a somewhat trivial question). Do prime numbers have prime factors, i.e. itself? For example is 7 a prime factor of 7?

The reason I ask this is because there is a statement in my lecture notes: If a number $n>1$ is not prime, then it has a prime factor.

I was hoping to restate this more generally as, all integers $n>1$ have prime factors.

Also, does it make more semantic sense to say that a prime $p$ is an integer, a natural number (i.e. $p\in \{0,1,2...\})$, or a positive integer?

Answer 1

This statement from your lecture notes

If a number $n>1$ is not prime, then it has a prime factor.

is true - but it's not a very good way to say what it's trying to say.

Here's an expanded version.

Every integer $n > 1$ has a prime factor. If $n$ happens to be prime then that prime factor is $n$ itself. If $n$ is not prime then it has a prime factor less than itself.

For the last part of your question: every prime is an integer, a natural number and a positive integer since every positive integer is also a natural number and an integer. It's probably best to use the most restrictive description - a prime is a positive integer - in fact, an integer greater than 1.

Answer 2

Yes, all integers $n>1$ have prime factors.

Composite numbers have prime factors less than themselves.

Prime numbers have no prime factors less than themselves.