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.

Answer 1

use the definition

Answer 2

\(a \equiv b \mod m\)

means

\(m | (a-b)\)

Answer 3

\(\large \implies a-b = mk\)

Answer 4

\(\large a = mk +b\)

substitute this value in the left side of congruence you wanted to prove

Answer 5

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

QED.

Answer 6

Hare Ganesh!

Answer 7

let me know if something is not clear :)