 Solution

A primitive Dirichlet character modulo $q$ is a Dirichlet character which is not induced from a character of a proper divisor of $q$. To determine the inducing primitive Dirichlet character, use the Chinese remainder theorem to decomposeOn each factor choose the least exponent $c_p\in \{0,\dots ,k\}$ through whose reduction the restricted character factors. Exponent zero means the trivial unit group modulo one. Put $q_0={\prod }_p{p}^{c_p}$ and define ${\chi }_0$ on the units modulo $q_0$ by these descended factors, extending by zero off the units. Every reduction of unit groups is surjective, so the descended character is unique. Its local exponents cannot be decreased, hence it is primitive. The original character is $\chi (n)={\chi }_0(n)$ when $(n,q)=1$, and zero otherwise.

Any other inducing modulus must have exponent at least $c_p$ at every prime, by restriction to the corresponding local factor. Therefore $q_0$ is the unique minimal modulus, the conductor of a Dirichlet character, and ${\chi }_0$ is the unique primitive Dirichlet character inducing $\chi$. The argument also explains why removing extra prime factors can change values at integers that were nonunits for $q$.