simplyfing modular equations

Question

Are there other ways of symplifing equations?

1. $\qquad$ Finding the inverse:

$ 42 y\equiv1 \bmod 5 \qquad 42 =5(8)+2 \qquad 5 = 2(2) +1 \qquad 2=1(2) \implies 1 = (17)5+42(-2) \implies x \equiv -2 \bmod 5 $

This method works because of Bezout’s identinty and Euclidean alghorithm

2. $\qquad 42 y\equiv1\bmod5 \qquad$ we know that $ 42 = 40 +2= 5(8)+2 \implies y \equiv2 \bmod5$

3. $\qquad$ Substracting the module

$ x\equiv 417 \bmod 120 \implies x \equiv 417 - 120 (3) \bmod120 \implies x \equiv 57 \bmod 120$

Are there other methods to simplify equations?