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(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?

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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.

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  • $\begingroup$ Except primes exist in all euclidean domains $\endgroup$ – Zelos Malum Sep 23 '16 at 2:47
  • $\begingroup$ @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. $\endgroup$ – Ethan Bolker Sep 23 '16 at 12:42
  • $\begingroup$ 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 $\endgroup$ – Zelos Malum Sep 23 '16 at 13:12
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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.

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