# Basic modular arithmetic proofs

Prove that congruence modulo $$m$$ is reflexive:

$$a \equiv a \pmod{m}$$

Prove that congruence modulo $$m$$ is symmetric:

$$a \equiv b \pmod{m} \longrightarrow b \equiv a \pmod{m}$$

Prove that congruence modulo $$m$$ is transitive:

$$a \equiv b \pmod{m} \wedge b \equiv c \pmod{m} \longrightarrow a \equiv c \pmod{m}$$
$$a \equiv b \pmod{m} \wedge c \equiv d \pmod{m} \longrightarrow a + c \equiv b + d \pmod{m}$$
$$a \equiv b \pmod{m} \wedge c \equiv d \pmod{m} \longrightarrow ac \equiv bd \pmod{m}$$
Let $$n \in \mathbb{Z}$$. Then $$n^2 \equiv 0$$ or $$1 \pmod{4}$$.
Let $$n$$ be a positive odd integer. Then $$n^2 \equiv 1 \pmod{8}$$.
Let $$m$$ be a natural number, and let $$x$$ and $$y$$ be integers. If $$x \equiv y \pmod{m}$$, then $$x$$ and $$y$$ have the same remainder upon division by $$m$$.
Let $$n \in \mathbb{N}$$. Then $$8 \mid 5^{2n} - 1$$.