The product of any nonzero irrational number and any integer constant is irrational.

Question

Can anyone help me find the counterexample to this problem?

Answer 1

If with "find the counterexample" you mean "find a non-zero irrational number $x$ and an integer $n$ such that $x\cdot n$ is not irrational", then note that $0$ is an integer, and for any $x$ we have $x\cdot 0=0$.

It's also easy to see that $0$ is the only integer with this property, since for any other integer $n$, if $x\cdot n = \frac{p}{q}$ with $p$ and $q$ integers, we'd have $x=\frac{p}{q\cdot n}$, a rational number.