# Do prime numbers have prime factors?

(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?

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.

• Except primes exist in all euclidean domains Sep 23, 2016 at 2:47
• @ZelosMalum Yes, and there are primes in other rings too. But that level of abstraction is probably not useful for the OP at his level of learning. Sep 23, 2016 at 12:42
• I think it is good to mention others at the end just to make them see there is much more but not necciserily be detailed Sep 23, 2016 at 13:12

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.