Mathematics of remainders.
In the following we are only talking about integers: Positive, Negative, and zero.

Last Up date on: 2004 August 9
Started: Sun 04-08-08 22:34:36

This an introduction to Congruence Arithmetic. I am using the notation and following what Carl Friedrich Gauss published in 1801.

Gauss' Definition:

Two integers a and b shall be said to be congruent for the modulus m when their difference a - b is divisible by the integer m.

Gauss expressed this in the symbolic statement

a ≡ b (mod m)

From the definition we have:

a - b = km
or
a = b + km
An alternate definition could be:
a and b are congruent modulus m if they have the same remainder when divided by m.
We will prove this later!
Simple observations from Gauss' definition:
  1. For any pair integers, either they are congruent for mod m or they are not.
  2. a ≡ a (mod m)
    because a - a = 0(m) for any m
  3. If a ≡ b (mod m) then b ≡ a (mod m)
    because if a - b = km then b - a = -km
  4. If a ≡ b (mod m) and b ≡ c (mod m)
    then a ≡ c (mod m)
    observe a - c = (a - b) + (b - c)
  5. Note: A precise way to write a is divisible by m is to write:
    a ≡ 0 (mod m) it makes it harder for the uninitiated to read! The sort of thing Mathematicians like to do!

Write several numeric examples, and observe the above "rules".
When, you are sure you are competent: go to the next page.

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