Answer to Question #154875 in Calculus for krishan k

Question

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

Answer 1

A number is said to be rational number if it is expressed as $\frac{p}{q}$ form, where $q\neq0$ and $g.c.d$ $(p,q)=1$ .

Now to prove the statement two cases arises :

Case 1: If $p,q$ have no common factor then $g.c.d$ $(p,q)$ =1.which is obvious.

Case 2: If $p,q$ have common factor.

Let us take $r$ is a common factor of $p$ and $q$ .

Then $p=r.m$ for some $m\in Z$

$q= r.n$ for some $n\in Z-${0} [ since $q\neq0,$ then $n\neq0$ ]

and $g.c.d$ $(m,n)=1$

In this case $\frac{p}{q}$ can be written in the form $\frac{m}{n}$, where $g.c.d$ $(m,n)=1$ . Which is also in the form of $\frac{p}{q}$ .

Therefore from these two cases we can conclude that every rational number can be expressed as a quotient of $\frac{p}{q}$ , so that $p$ and $q$ have no common factors.