# Congruence of Sum with Constant

From ProofWiki

## 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 |

$\blacksquare$