In both cases, it helps to consider the Lucas numbers modulo $m$. For example, modulo $3$, the Lucas numbers (zero-indexed) begin $$2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, \dots$$ and you may be able to spot a periodic pattern here: the sequence $2,1,0,1,1,2,0,2$ repeats over and over.

If you prove this periodic pattern, then your first statement follows just by looking where the $0$ appears. Similarly, if you find and prove a periodic pattern for $L_k \bmod 4$, then the second statement will follow by looking where the $3$ appears.

The pattern can be proved by some kind of induction. Essentially, knowing $L_k$ and $L_{k+1}$ modulo $m$ tells you $L_{k+2}$ modulo $m$, so once the mod-$m$ sequence repeats its starting values of $2,1$ once, you know that it will repeat them forever.