## Problem 4.

Let $m \geq 1$ be a positive integer, and $a$ and $b$ be distinct positive integers strictly greater than $m^{2}$ and strictly less than $m^{2}+m$. Find all integers $d$, that divide the product $ab$ and satisfy $m^{2}<d<m^{2}+m$.
## Solution.

Let $d$ be a positive integer that divides $ab$ and such that $d \in (m^2, m^2 + m)$. Then $d$ divides $(a-d)(b-d) = ab - da - db + d^2$. Since $|a-d| < m$ and $|b-d| < m$, we deduce that $|(a-d)(b-d)| < m^2 < d$, which implies that $(a-d)(b-d) = 0$. Thus, $d = a$ or $d = b$.