KöMaL 2024 B5405

Positive integers $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ satisfy the following property: for all indices $i < j \leq n$ the greatest common divisor of $b_i$ and $b_j$ does not divide difference $(a_i-a_j)$. Prove that $\sum_{i=1}^n \frac1{b_i} \leq 1$.