I [Congruence class] Proof of modular arithmetic theorem

[Leo Liu]

TL;DR Summary

.

Could someone explain why $[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)$?
My instructor said it came from the definition of congruence class. But I am not convinced.

 

[ergospherical]

Recall that $[x] = [y]$ if $x \equiv y \ (\mathrm{mod \ m})$. Then it's just a case of using $[a][x_0] = [ax_0]$.

 

[Leo Liu]

Recall that $[x] = [y]$ if $x \equiv y \ (\mathrm{mod \ m})$. Then it's just a case of using $[a][x_0] = [ax_0]$.

Wonderful. I didn't know this form of definition but after seeing its justification (transitivity and symmetry) I am convinced.