Assume p|mn, thus for some integer q, pq=mn. By definition of prime, the gcd(m,p)=1 or gcd(m,p)=p. If gcd(m,p)=p, then p|m and the desired result is shown. If gcd(m,p)=1, then p|gcd(m,p)n. See here for a proof.
Here is the whole proof I outlined above. I'm sorry to link so much but I don't really know what you can use or not so I don't want to type out a bunch of stuff for nothing. Does the above info help?
But nonetheless, strictly speaking it is incorrect.
That definition makes 1 a prime number.
The number 1 has only 1 and itself as divisors.
So I think that the correct wording is: A positive integer is prime if and only if it has exactly two divisors.
You understand that 1 has only one divisor: itself.
I don't know if you can use this though, or if you are supposed to do it another way.