More Congruence Rules
In the following we are only talking about integers: Positive, Negative, and zero.

Last Up date: 2005 October 17
Started: Mon 04-08-09 11:49:12

Still from Gauss.

Multiplication Rule
If a ≡ b (mod m) and c ≡ d (mod m)
then ac ≡ bd (mod m)
Observe we can multiply the first by c to get:
ac ≡ bc (mod m)
and multiply the second by b and get:
bc ≡ bd (mod m)
Apply rule 4 on page 1. QED

By repeated application of the multiplication rule it follows:
One can multiply an arbitrary set of congruences for the same modulus.

In particular a congruence may be multiplied by itself any number of times so we have:

If a ≡ b (mod m)
then ak ≡ bk (mod m)

A congruence may be raised to an arbitrary positive integer power.

Since any of the operations: Addition, Subtraction, and Multiplication when applied to congruent numbers will give congruent results, we conclude that any algebraic expression constructed by use of these operations will give congruent results when congruent values are substituted.

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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