Chef and Working Plan

These days, Chef is working on opening a restaurant. Obviously, Chef dislikes running the restaurant by himself. In order to avoid such a terrible thing, he decided to hire exactly $M$ workers (numbered $1$ through $M$) to help him. The workers will choose some positive integer constants $a_1, a_2, \dots, a_M$. A worker with a constant $a$ wants to be paid a fixed salary after each $a$ days. Ch