Congruence of Sum with Constant
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Theorem
Let $a, b, z \in \R$.
Let $a$ be congruent to $b$ modulo $z$:
- $a \equiv b \pmod z$
Then:
- $\forall c \in \R: a + c \equiv b + c \pmod z$
Proof
Follows directly from the definition of Modulo Addition:
\(\ds a\) | \(\equiv\) | \(\ds b\) | \(\ds \pmod z\) | given | ||||||||||
\(\ds c\) | \(\equiv\) | \(\ds c\) | \(\ds \pmod z\) | Congruence Modulo Real Number is Equivalence Relation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds a + c\) | \(\equiv\) | \(\ds b + c\) | \(\ds \pmod z\) | Definition of Modulo Addition |
$\blacksquare$