Let be a Dirichlet character mod and let be any positive divisor of . The number is called an induced modulus of if we have
whenever and (mod )
I wanted to construct an intuitively straightforward proof for myself that if is a character mod , and if and are induced moduli for , then must also be an induced modulus for , and . I managed to come up with a relatively simple proof as follows.
Suppose (mod ), with . Since and are induced moduli for , must take equal values at numbers which are relatively prime to and congruent modulo either or . In other words we must have
for any integers such that
This is only possible in general if takes equal values at numbers which are relatively prime to and congruent modulo some common divisor of and (otherwise in some cases it would be possible to find some such that ). Since any common divisor of and must also divide their greatest common divisor , it follows that must take equal values at numbers which are relatively prime to and congruent modulo . Thus, since (mod ), we have
and is an induced modulus for if and are. QED