Problem

Source: Iran 3rd round 2011-final exam-p8

Tags: algebra proposed, algebra



We call the sequence d1,....,dn of natural numbers, not necessarily distinct, covering if there exists arithmetic progressions like c1+kd1,....,cn+kdn 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 d1,....,dn such that the resulting sequence be still covering. a) Suppose that d1,....,dn is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions a1+kd1,.....,an+kdn, and suppose that p is a prime number that p divides d1,....,dk but it does not divide dk+1,....,dn. Prove that the remainders of a1,....,ak modulo p contains all the numbers 0,1,.....,p1. b) Write anything you can about covering sequences and short covering sequences in the case that each of d1,....,dn has only one prime divisor. proposed by Ali Khezeli