If $n \in \mathbb{Z}^+$ and $x \in \mathbb{R}$, show that $x \in \mathbb{Q}$ if $n x \in \mathbb{Q}$

Question

For this question would I be correct in stating the following

If $n$ is a positive integer and $n x$ is a rational number then $n x = n (a / b)$.

Simplifying this = $x = (a / b)$ so $x$ must also be a rational number.

I'm not entirely sure if making the first claim is correct.

Answer 1

The correct argument is as follows: $nx=\frac a b$ with $a,b$ intergers, $b \neq 0$. Hence $x= \frac {a} {nb}$ which is rational. It is not logical to assume that $nx=n\frac a b$.

Answer 2

No. This argument is not valid.

If $nx$ is rational then $nx = \frac ab$ for integers $a,b$.

The only way that $nx = n \frac ab$ would be if $x = \frac ab$ which is what you are trying to prove.

You are assuming what you are trying to prove. That is a no-no and is always wrong.

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But if you start with $nx =\frac ab$ then the proof pretty much writes itself. $x = \frac a{bn}$ and so....