Congruence of Sum with Constant

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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:

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

$\blacksquare$