Product of 2 coprime $a$ and $b$ both less than $n$ and both coprime to $n$ can never be $xn$ where $x$ is any positive integer

Question

I am having trouble proving that product of two coprime $a$ and $b$ can never generate a number multiple of some greater number which is coprime to both $a$ and $b$.

Answer 1

We know that $n\mid xn.$ For $ab=xn$ , it follows that $n\mid ab$ . But we are given that $n\not\mid a,b$ , a contradiction.

Hence for coprimes $a,b$ it follows that $ab \ne xn.$

Answer 2

Suppose that $ab=xn$, with your notation.

As $a$ is coprime with $n$ and $a|xn$, $a|x$, because $a$ or any divisor of $a$ cannot divide n. The same reasoning applies to $b$.

But now we have $ab=xn$ and $ab|x$. Thus, $ab=x$ and $n=1$, which is a contradiction (for example it would imply $a=b=0$).