Modulo Addition is Linear
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Theorem
Let $m \in \Z_{> 0}$.
Let $x_1, x_2, y_1, y_2, c_1, c_2 \in \Z$.
Let:
- $x_1 \equiv y_1 \pmod m$
- $x_2 \equiv y_2 \pmod m$
Then:
- $c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$
Proof
By Scaling preserves Modulo Addition:
- $c_1 x_1 \equiv c_1 y_1 \pmod m$
- $c_2 x_2 \equiv c_2 y_2 \pmod m$
and so by Modulo Addition is Well-Defined:
- $c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$
$\blacksquare$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)} \ \text{(A)}$