Let $a$, $b$ be the positive integers greater than $1$. Prove that if $$ \frac{a}{b},\; \frac{a-1}{b-1} $$differ by 1, then both are integers.
Source: Polish Junior Math Olympiad 2019 Finals
Tags: number theory
Let $a$, $b$ be the positive integers greater than $1$. Prove that if $$ \frac{a}{b},\; \frac{a-1}{b-1} $$differ by 1, then both are integers.