Question

Exercise 1. Every rational number can be expressed as a quotient p/q so that p and q has no common factors.

Answer 1

By definition, a rational number $x$ is a quotient of two integers, we will note them as $a, b$, where $b\neq 0$. If $a=0$, then $x=0 = \frac{0}{1}$ is convenient. If $a\neq0,$ we note $r=gcm(a,b)\neq0$, we have $x = \frac{a}{b}= \frac{a/r}{b/r}$, where $a/r, b/r$ are integers (as $r|a, r|b$), $b/r\neq 0$ and $gcm(a/r, b/r)=1$ (as if it was not the case, that would contradict to the fact that $r$ is the greatest common divisor). Therefore $p=a/r,q= b/r$ are integers that have no common factors and $x=\frac{a}{b}= \frac{a/r}{b/r}=\frac{p}{q}$