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 \(8 \mid 5^{2n} - 1\).