Find the Kernel of $\Psi : (\Bbb{Z}_{30},+_{30})\rightarrow (\Bbb{Z}_{20},+_{20})$

Question

Find the Kernel of $\Psi : (\Bbb{Z}_{30},+_{30})\rightarrow (\Bbb{Z}_{20},+_{20})$

Provided that $\Psi([b])=[4b]$

My understanding of the kernel is that it should be:

$\ker(\Psi)=\{[b]\in\Bbb{Z}_{30} | [4b]_{20}=[0]_{20}\}$

Then, you can see that $b=0,5,10,15,20,25$? Is this correct?

My professor had written

$b=0,5,10,15$ but I do not see why, I am not sure if he was mistaken

Answer 1

Yes, all (equivalence classes of) multiples of five modulo $30$ will do the trick. Perhaps your professor confused the codomain with the domain.