Let m be a positive integer. Show that a mod m =
b mod m if a ≡ b (mod m). Use only the definitions of mod m and congruence modulo m; do
not use any theorems involving mod or congruence.

Question

Let m be a positive integer. Show that a mod m =
b mod m if a ≡ b (mod m). Use only the definitions of mod m and congruence modulo m; do 
not use any theorems involving mod or congruence.

ganeshie8 · Answer

use the definition

ganeshie8 · Answer

$a \equiv  b \mod m$

means

$m | (a-b)$

ganeshie8 · Answer

$\large    \implies   a-b =  mk$

ganeshie8 · Answer

$\large a = mk +b$

substitute this value in the left side of congruence you wanted to prove

ganeshie8 · Answer

$\large a \mod m   =   (mk + b) \mod m  \equiv  b \mod  m$

QED.

anonymous · Answer

Hare Ganesh!

ganeshie8 · Answer

let me know if something is not clear :)