Question 64
Let $m$ and $n$ be positive integers and let the integer $X \ge \max(m, n)$. Show that there exist integers $u$ and $v$, not both equal to $0$, such that
$$
\max(|u|, |v|) \le \sqrt{X} \quad \text{and} \quad 0 \le m u + n v \le 2\sqrt{X}.
$$
$$
\max(|u|, |v|) \le \sqrt{X} \quad \text{and} \quad 0 \le m u + n v \le 2\sqrt{X}.
$$