MathNet 2019-part-03 — 184 problems (EN–VI) Question 100

Let $n$ and $M$ be positive integers such that $M \gt n^{n-1}$. Prove that there are $n$ distinct primes $p_{1}, p_{2}, p_{3}, \ldots, p_{n}$ such that $p_{j}$ divides $M + j$ for $1 \leq j \leq n$.