0

Saying a is equivalent b mod m is like saying:

a = b - qm

where q is the quotient, m is the divisor, and a will equal the remainder. ^ Verify the above statement

So then if we were to solve this equation for qm we would get:

a-b=-qm

This logic is contradicted by my professor when looking at one of his proof examples where he states a = b mod m implies a q exists such that:

a-b=qm

The contradiction is qm is not negative. I understand this has something to do with congruence, but I am not quite understanding how this is correct and where my logic is flawed.

Community
  • 1
Mathwell
  • 61
  • 4
    The sign of $q$ is irrelevant. Writing $a\equiv b\pmod m$ is the same as writing $b\equiv a \pmod m$. – lulu Jun 28 '19 at 11:17
  • It is as if your naming a new number $q'=-q$ and then writing $a-b=q'm$ – Fareed Abi Farraj Jun 28 '19 at 12:08
  • Congruence (residue) arithmetic focuses on the arithmetic of residue classes (remainders) modulo $m$. The quotients $q$ play no essential role in this arithmetic. The sooner you learn to forget about the quotients the quicker you will be able to grasp the essence of the matter. – Bill Dubuque Jun 28 '19 at 12:22
  • a = b mod m means that a - b is divisible by m. It is a relation between two numbers. b mod m is not an operation between two numbers as programmers delude themselves. – William Elliot Jun 28 '19 at 12:45
  • @WilliamElliot Both the congruence relation $\ a\equiv b\pmod{m}\ $ and remainder operation $\ b \bmod m\ $ are in wide use (in both Math & CS). Which proves best depends on the context. See here for more. – Bill Dubuque Jun 28 '19 at 20:48

0 Answers0