The largest one of numbers p1α1,p2α2,⋯,ptαt is called a <spanclass=′latex−bold′>GoodNumber</span> of positive integer n, if \displaystyle n\equal{} p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t}, where p1, p2, ⋯, pt are pairwisely different primes and α1,α2,⋯,αt are positive integers. Let n1,n2,⋯,n10000 be 10000 distinct positive integers such that the <spanclass=′latex−bold′>GoodNumbers</span> of n1,n2,⋯,n10000 are all equal.
Prove that there exist integers a1,a2,⋯,a10000 such that any two of the following 10000 arithmetical progressions \{ a_i, a_i \plus{} n_i, a_i \plus{} 2n_i, a_i \plus{} 3n_i, \cdots \}( i\equal{}1,2, \cdots 10000) have no common terms.