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:

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


Proof

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          

$\blacksquare$