Congruence of Sum with Constant

From ProofWiki

Jump to: navigation, search

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

$\blacksquare$

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