Solved
I am self studying analytic number theory from Tom M Apostol and got stuck on this problem on page176.
It's image:
Problem is in question 10 only.
I have proved that $f(\chi) $divides $f(\chi_{1})... f(\chi_{r}) $ but I am unable to think about how can I prove the converse.
can you please help with that?
Edit : Smallest induced modulus is called conductor and Induced Modulus: Let $\chi$ be a Dirichlet Character mod k and let d be any positive divisor of k. The number d is called an induced modulus for $\chi$ if we have $\chi$(a)=1 whenever 9a,k)=1 and a$\equiv $1 (mod d). ( Page 167 of the book.)