Valentin Vornicu wrote: The set of positive integers is partitionated in subsets with infinite elements each. The question (in each of the following cases) is if there exists a subset in the partition such that any positive integer has a multiple in this subset. a) Prove that if the number of subsets in the partition is finite the answer is yes. b) Prove that if the number of subsets in the partition is infinite, then the answer can be no (for a certain partition). a) suppose au contraire that the subset $X_i$ in the partition lacks the multiples of $n_i$ for every subset. In that case. where is the number $n_1n_2...n_k$? b) The following partition will do. $X_n = \{ n^k, k = 1,2,.../ n \neq a^m, m>1 \}$ Daniel

a) Some set gets infinte numbers of the form $n!$.