0

I need to prove that each of the congruence classes listed above are different, and that any other congruence class [a](for any integer a) has to be equal to one of the congruence classes listed.

I know in order to prove the second part I must show that a is a subset of a congruence class and that congruence class is a subset of a to show they're equal but I'm not sure how to go about the first part.

Hannnnn
  • 35

1 Answers1

0

The congruence class of an integer is the congruence class of its remainder when divided by $m$. Two remainders can't be congruent, since they're their own remainders. What is the list of the possibles remainders?

Bernard
  • 175,478
  • I need to show that each congruence class in {[0], [1], ..., [m-1]} are actually different – Hannnnn Mar 27 '17 at 23:08
  • Remainder in Euclidean division is unique. Thus $[i]\ne [j]$ if $;0\le i,j<m$ and $i\ne j$. – Bernard Mar 27 '17 at 23:10
  • i still dont understand how to prove that each congruence class is different – Hannnnn Mar 27 '17 at 23:20
  • Two integers are in the same congruence class if they have the same remainder when divided by $m$. Do two distinct natural numbers less than $m$ have the same remainder? – Bernard Mar 27 '17 at 23:28