Are you by any chance using Herbert Endertonís text ELEMENTS OF SET THEORY?
If so, this problem appears in it with a nice theorem about function compatible with equivalence relations.
If not, here is the basic idea. Consider the function .
It should be clear to you that if have the same number of prime factor then so do .
In other words, this is enough to say is compatible with .
Thus there exists .
If you need details your library should a copy of that text.