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I'm taking 'Introduction to Mathematical Thinking' on Coursera platform and following proof steps are given :

Proof of $\sqrt{2}$ is irrational.

Assume $\sqrt{2}$ is rational. $\sqrt{2}=p/q$

p and q have no common factors.

Why do p and q have no common factors? Is this a consequence of a property of the rational numbers? As p and q can be rational numbers we can set p = 6, q = 9 so p, q have common factors?

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    You know that if $x$ is rational, then there exist two numbers $p,q$ such that $p$ and $q$ have no common factors and $x=\frac pq$. – 2017-02-09

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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.

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    is 1 excluded from common factors as p,q share 1 as common factor ? – 2017-02-09
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    @blue-sky yes: if 1 is the only common factor, then the numbers are said to have no common factors. – 2017-02-09
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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 $\sqrt{2} \in \mathbb Q$