IR-MO 2011 Problem 8

We call the sequence $d_1,....,d_n$ of natural numbers, not necessarily distinct, covering if there exists arithmetic progressions like $c_1+kd_1$ ,...., $c_n+kd_n$ such that every natural number has come in at least one of them. We call this sequence short if we can not delete any of the $d_1,....,d_n$ such that the resulting sequence be still covering.a) Suppose that $d_1,....,d_n$ is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions $a_1+kd_1,.....,a_n+kd_n$ , and suppose that $p$ is a prime number that $p$ divides $d_1,....,d_k$ but it does not divide $d_{k+1},....,d_n$ . Prove that the remainders of $a_1,....,a_k$ modulo $p$ contains all the numbers $0,1,.....,p-1$ .b) Write anything you can about covering sequences and short covering sequences in the case that each of $d_1,....,d_n$ has only one prime divisor.

proposed by Ali Khezeli