lean_workbook_plus_58008

Prove that there exists a number $t$ such that $t, t+n_i$ $(i \in (1, 2, ..., r))$ are not divisible by $p_i$ for all $i \in (1, 2, ..., k)$ given a set of finite primes $P = (p_1, p_2, ..., p_k)$ and numbers $n_1, n_2, ..., n_r$ which are not divisible by any $p_i \in P$.