Question 68
Given $m, n \in \mathbb{N}$ such that $m \gt n^{n-1}$ and the numbers $m+1, m+2, \dots, m+n$ are composite.
Prove that there exist distinct primes $p_1, p_2, \dots, p_n$ such that $m+k$ is divisible by $p_k$ for any $k = 1, 2, \dots, n$.
Editorial metadata note: Metadata discrepancy: the dataset year is 2014, while booklet_year is 2015. The title follows dataset classification, not a verified Belarus 2014 event date.
Prove that there exist distinct primes $p_1, p_2, \dots, p_n$ such that $m+k$ is divisible by $p_k$ for any $k = 1, 2, \dots, n$.
Editorial metadata note: Metadata discrepancy: the dataset year is 2014, while booklet_year is 2015. The title follows dataset classification, not a verified Belarus 2014 event date.