Why no common factors in proving root 2 is irrational?

Solution 1:

This is because every rational of the form $$\frac{a}{b}$$ can be simplified to the form $$\frac{p}{q}$$ where $p$ and $q$ are coprime. This follows from the fact that if $\gcd(a,b)=d$ then $a=pd, b=qd$ where $p$ and $q$ are copime as seen here from the property of the common divisor.

So we are trying to express $\sqrt{2}$ in the simplest way possible, which should always be possible if it is a rational.

Solution 2:

If you assume that $\sqrt{2}=p/q$ with $p,q \in \mathbb N$, then the usual proof for " $\sqrt{2}$ irrational" shows that $p$ and $q$ has the common factor $2$.

If you start(!) the proof with $\sqrt{2}=p/q$ and $p$ and $q$ have no common factors (which is possible), then you get a contradiction, which shows that we can not have $\sqrt{2} \in \mathbb Q$