$a \equiv b \pmod{n}$ and $c$ is an integer. Which of the following are always true?

I. $a + c \equiv b + c \pmod{n}$

II. $ac \equiv bc \pmod{n}$

III. $a \equiv b \pmod{2n}$

IV. $a^{2} \equiv b^{2} \pmod{n}$

Explanation

1. 01

   Adding, multiplying and raising to a power keep a congruence: if $n$ divides $a - b$, it divides $(a - b)c$ and $a^{2} - b^{2} = (a - b)(a + b)$.

2. 02

   III fails: $1 \equiv 4 \pmod{3}$, but 6 does not divide $4 - 1 = 3$.