Question

Every rational number can be expressed as a quotient p/q so that p and q has no common factors.

Answer 1

Let the rational number be $a=\frac{p}{q}$ , such that $p,q\in\Z$ and gcd($p,q$)=1

Let us assume that gcd($p,q$)$\neq1$ or gcd($p,q$)=c(let), such that c$\in\Z$

Then we write a as:-

$$a=\frac{p}{q}=\frac{\frac{p}{c}}{\frac{q}{c}}=\frac{p'}{q'}$$

where, $cp'=p$ and $cq'=q$ , where $p',q'\in\Z$ as $p,q,c\in\Z$

Then,

$$\Rightarrow a=\frac{p'}{q'}$$

where, $p',q'\in\Z$ and gcd($p',q'$)=1

Therefore, our assumption that gcd($p,q$)=c still holds the definition of rational number.