Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA
Source: 2015 Belarus TST 8.1
Tags: number theory, primes, divisible, consecutive
Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA