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Ok,the way I'm thinking about it is using a proof by contradiction:asuming ker f=12Z....then by the First Isomorphism Theorem for rings Z/ker f ~im f where im f is by definition a subring of C.But since im f=12Z is not an integral domain and every subring in C is an integral domain the im f will not be a subring oc C which is a contradiction.

The same thing with 13Z,is not equal with the kernel.