Proof of Modular Arithmetic Addition

Proof

WTS: $((a\mathrm{mod}n)+(b\mathrm{mod}n))\mathrm{mod}n=(a+b)\mathrm{mod}n$

With $a,b\in \mathbb{Z}$, then $a=q_{1}n+r_1$, $b=q_{2}n+r_2$

LS

Consider $(a\mathrm{mod}n)+(b\mathrm{mod}n)\mathrm{mod}n=(r_1+r_2)\mathrm{mod}n$

RS

Consider $(a+b)\mathrm{mod}n=((q_1+q_2)n+r_1+r_2)\mathrm{mod}n=(r_1+r_2)\mathrm{mod}n$ Thus, LS = RS