1. [2] Suppose $x$ is a rational number such that $x \sqrt{2}$ is also rational. Find $x$.
Answer: $\square$
Suppose $x \neq 0$. Then $\frac{x \sqrt{2}}{x}=\sqrt{2}$ is the quotient of two nonzero rationals, and so is rational. However, it is well-known that $\sqrt{2}$ is irrational. Therefore any solution $x$ must satisfy $x=0$. We can see that 0 is rational and $0 \sqrt{2}=0$ is rational, so the answer is indeed $x=0$.