# Congruence of Sum with Constant

## Theorem

Let $a, b, z \in \R$.

Let $a$ be congruent to $b$ modulo $z$, i.e. $a \equiv b \ \left({\bmod\, z}\right)$.

Then:

$\forall c \in \R: a + c \equiv b + c \ \left({\bmod\, z}\right)$.

## Proof

Follows directly from the definition of Modulo Addition:

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle a$$ $$\equiv$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle b$$ $$\displaystyle \pmod z$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle c$$ $$\equiv$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle c$$ $$\displaystyle \pmod z$$ $$\displaystyle$$ as Congruence (Number Theory) is Equivalence Relation $$\displaystyle$$ $$\displaystyle \implies$$ $$\displaystyle$$ $$\displaystyle a + c$$ $$\equiv$$ $$\displaystyle$$  $$\displaystyle$$ $$\displaystyle b + c$$ $$\displaystyle \pmod z$$ $$\displaystyle$$ Modulo Addition

$\blacksquare$