Number Theory/Congruences

= Notation and introduction =

We will call two integers a and b congruent modulo a positive integer m, if a and b have the same (smallest nonnegative) remainder when dividing by m. The formal definition is as follows.

Definition
Let a, b and m be integers where $$m>0$$. The numbers a and b are congruent modulo m, in symbols $$a \equiv b \pmod{m}$$, if m divides the difference $$a-b$$.

Lemma
We have $$a \equiv b \pmod{m}$$ if and only if a and b have the same smallest nonnegative remainder when dividing by m.

Proof:

Let $$a \equiv b \pmod{m}$$. Then there exists an integer c such that $$cm = a-b\,$$. Let now $$q,q',r,r'\,$$ be those integers with

$$a=qm+r,\quad  0\leq r<m$$

and

$$b=q'm+r',\quad 0\leq r'<m$$.

It follows that

$$cm = a-b = m(q-q') + (r-r')\,$$

which yields $$m|(r-r')\,$$ or $$m\ \mbox{divides} \ (r-r')\,$$ and hence $$r=r'\,$$.

Suppose now that $$r=r'\,$$. Then, $$a-b = m(q-q')\,$$, which shows that $$m|(a-b)\,$$.

= Fundamental Properties =

First, if $$a\equiv b \pmod{m}$$ and $$c\equiv d \pmod{m}$$, we get $$ac\equiv bd \pmod{m}$$, and $$a+c\equiv b+d \pmod{m}$$.

As a result, if $$a\equiv b \pmod{m}$$, then $$ a^p\equiv b^p \pmod{m}$$

= Congruence Equations =

= Residue Systems =

Chinese Remainder Theorem
= Polynomial Congruences =