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:

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

Follows directly from the definition of Modulo Addition:

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

$\blacksquare$

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