Equal Numbers are Congruent

Theorem

$\forall x, y, z \in \R: x = y \implies x \equiv y \pmod z$

where $x \equiv y \pmod z$ denotes congruence modulo $z$.

Proof

\(\ds x\)

\(=\)

\(\ds y\)

\(\ds \leadsto \ \ \)

\(\ds x - y\)

\(=\)

\(\ds 0\)

\(\ds \leadsto \ \ \)

\(\ds x - y\)

\(=\)

\(\ds 0 \cdot z\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\equiv\)

\(\ds y\)

\(\ds \pmod z\)

Definition of Congruence Modulo $z$

$\blacksquare$

Also see

• Congruent Numbers are not necessarily Equal

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Example $\text {4-3}$