Modulo Addition is Linear

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: $\text{(iii)} \ \text{(A)}$