Create a set of N numbers with no common rational factor Question:
So I want to create a set of real numbers $\{a\}_{N} = \{a_{1}, a_{2}, \ldots, a_{N}\}$ such that if there exists a common factor between all of the elements, it must be irrational. 
In this way, the fraction $\frac{a_{i}}{a_{j}} \notin \mathbb{Q} \; \forall i \neq j \; \; (1 \leq i,j \leq N)$.
How can I create this set?

Solution Attempt:
We know that the square root of any prime number is irrational, therefore, just pick the set $\{a\}_{N}$ to be a set of square roots of distinct prime numbers.  
What is tripping me up is that, the division of two irrational numbers can still be rational. An easy example is $\frac{2 \sqrt{2}}{3\sqrt{2}} = \frac{2}{3} \in \mathbb{Q}$ 

Disclaimer:
I am not a number theorist, and have never taken a course on number theory, but I converted another problem I am working on to this problem. If I can solve this problem, I can solve the other problem, however, I don't know if this problem has a solution, and if it does, how to find it.

Well, your solution attempt also works. It's not that much different or more difficult to prove that $\frac{\sqrt{p}}{\sqrt{q}}$ is irrational for different primes $p,q$ than proving $\sqrt{2}$ is irrational. 
Your 'tripping me up' fear is not without reason, of course, but you circumvented it by chosing roots of primes, not just any number that isn't a square.
