Školjka

%V0 Let $P_1, \ldots , P_s$ be arithmetic progressions of integers, the following conditions being satisfied: (i) each integer belongs to at least one of them; (ii) each progression contains a number which does not belong to other progressions. Denote by $n$ the least common multiple of the ratios of these progressions; let $n=p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ its prime factorization. Prove that $$s \geq 1 + \sum^k_{i=1} \alpha_i (p_i - 1).$$ Proposed by Dierk Schleicher, Germany