Number Theory - Congruence implication proof

Question

How do you prove that a congruence (such as this) implies another? I'm not sure where to begin and I think I'm missing something simple.

$$ a \equiv b \ \text {mod} \ m \implies 4a \equiv 4b \ \text{mod} \ m $$

Answer 1

1. $a\equiv b \pmod{m}$. This implies
   
2. $m|(b-a)$. This implies
   
3. There is some integer $k$ such that $mk=b-a$. This implies
   
4. There is some integer $4k$ such that $m(4k)=4b-4a$. This implies
   
5. $m|(4b-4a)$. This implies
   
6. $4a\equiv 4b \pmod{m}$.