Problem

Source: 2024 Mexican Mathematics Olympiad, Problem 2

Tags: number theory, Mexico, Divisors



Determine all pairs $(a, b)$ of integers that satisfy both: 1. $5 \leq b < a$ 2. There exists a natural number $n$ such that the numbers $\frac{a}{b}$ and $a-b$ are consecutive divisors of $n$, in that order. Note: Two positive integers $x, y$ are consecutive divisors of $m$, in that order, if there is no divisor $d$ of $m$ such that $x < d < y$.