Modulo division: Find all integers $k \geq 2$ such that $k^2 = 5k(\mod 15).$

Question

Find all integers $k \geq 2$ such that $k^2 = 5k(\mod 15).$

Using arithmetic on $\mathbb{Z}_{15},$ $\bar{k}^2 = \bar{5}\bar{k},$ may I divide both sides by $\bar{k}$ to arrive at $\bar{k} = \bar{5}?$

Answer 1

Only if $k$ has an inverse $m$, with $km=1$, so you multiply by the inverse.
Numbers that are multiples of 3 or 5 don't have inverses. That is because any multiple of them is still a multiple of 3 or 5, and so not 1.
So check multiples of 3 and 5 separately.
Note that if the equation were $k^2=2k$, solutions would be 0,2,5 and 12.