I agree that "prime" is important. But I am trying to explain that a number is just a number. Try to define a prime number without using the word "product" nor the word "multiplication." Now, if the math professors can't do this then there is a serious problem with all the hype about primes. What will need to happen is that educators change their language-- change the way they talk about the phenomenon. The phenomenon is due to short cut addition. So, it is not the numbers that are cool but the short cuts we come up with for faster addition (in other words: multiplication).
In the pure mathematical universe time is not a factor so addition is just as fast as multiplication. However, when humans try to add in their head, they impose a time constraint. This is why the earliest mathematicians came up with multiplication. It is just a short cut since we humans are stuck in time. So, beg your professor to explain a prime number in terms other than multiplication and "product". I bet they can't do it or they will come up with an excuse about how they need to leave the room.