How to solve this modulo equation using modulo properties?

Question

Equation: $[3*(k \mod 4)] \mod 4 = 3$

It's relatively easy to check the equation for the possible values of $k \mod 4$. Is there a more elegant way to calculate the solution, for example by using modulo properties?

Answer 1

To solve $3k\equiv3\pmod4$, multiply both sides by the inverse of $3\pmod 4$ (which is $3$):

$3k\equiv3\pmod4\implies 3^{-1}3k\equiv3^{-1}3\pmod4\implies k\equiv1\pmod 4$