Carl (or Karl) Friedrich Gauss, prince of mathematicians, thought that ``Mathematics is the queen of the sciences and number-theory the queen of mathematics.'' His Disquisitiones Arithmeticae was the book Dirichlet carried about like the Bible. This notion of congruence appears in the first chapter....
You're probably already familiar with modular arithmetic: this is the generalization of it. On the clock, 13 and 1 are the same thing (if we ignore pm and am - 25 and 1 work if you don't want to ignore am and pm!).
Definition 4.1: Let n be a fixed positive integer. Two integers a and b are said to be congruent modulo n, symbolized by
if n divides a-b; that is, provided a-b=kn for some integer k.
complete set of residues: a collection of n integers
forms a complete set of residues modulo n if
every integer is congruent modulo n to one and only one of the collection.
(For those of you who've had linear algebra, you can think of the collection as
a ``basis'' for all integers with respect to the operation of congruence).
Theorem 4.1: For arbitrary integers a and b,
if and only if a and b leave the same
nonnegative remainder when divided by n.
Theorem 4.2: Let n>1 be fixed and a, b, c, and d be arbitrary integers. Then the following properties hold:
-
-
If , then
. -
If and
,
then 
. -
If and
,
then 
, and
. -
If ,
then
, and
. -
If , then
for any positive integer k.
Note that the converse of Theorem 4.2(f) is false: for example
Half of the converse of Theorem 4.2(e) is also false, as indicated in Theorem 4.3:
Theorem 4.3:
If 
, then 
,
where 
.
Corollary 1: If and
, then .
Corollary 2: If 
(p prime), and p
does not divide c, then 
.
Unfortunately we cannot simply cancel without thought: for example, you can check that
but that
In fact, however, it is true that
Because all integers are congruent modulo 1, we generally assume that in a
formula 
, n>1.
Note that 
does not imply that a or b is 0 (mod n):
for example
but neither 5 nor 3 is 0 (mod 15). What we can say is that, if
and 
, then 
.