[freemind]

Let be a positive integer. Prove that every non-zero positive integer has a multiple in the sequence , .

[Fedor Petrov]

Strange problem. Let , where and are coprime, and divides for suitable positive integer . Without loss of generality, divides (we may take being large power of ). Take large prime such that is divisible by . The existence of such is well-known elementary case of the Dirichlet theorem, which may be obtained using the cyclotomic polynomials. Then take . Then by Fermat's little theorem we have , which is divisible by .