Question 103
$n \gt 3$ distinct positive integers are written on a board, all these numbers are smaller than $(n-1)! = 1 \cdot 2 \cdot \dots \cdot (n-1)$. For each pair $a \gt b$ of these numbers, Sergey found a partial quotient of $a$ by $b$ and wrote up the result into his notebook (e.g., if $a = 100$ and $b = 7$, then $100 = 14 \cdot 7 + 2$, so Sergey would write the number 14 into the notebook). Prove that the notebook contains two equal numbers.