The theory of numbers, by Robert D. Carmichael ...

CHAPTER III

ELEMENTARY PROPERTIES OF CONGRUENCES
~ i8. CONGRUENCES MODULO m
DEFINITIONS. If a and b are any two integers, positive
or zero or negative, whose difference is divisible by m, a and b
are said to be congruent modulo m, or congruent for the modulus
m, or congruent according to the modulus m. Each of the
numbers a and b is said to be a residue of the other.
To express the relation thus defined we may write
a=b+cm,
where c is an integer (positive or zero or negative). It is more
convenient, however, to use a special notation due to Gauss,
and to write
a —b mod m,
an expression which is read a is congruent to b modulo n, or
a is congruent to b for the modulus m, or a is congruent to b
according to the modulus m. This notation has the advantage
that it involves only the quantities which are essential to the
idea involved, whereas in the preceding expression we had the
irrelevant integer c. The Gaussian notation is of great value
and convenience in the study of the theory of divisibility.
In the present chapter we develop some of the fundamental
elementary properties of congruences. It will be seen that
many theorems concerning equations are likewise true of congruences with fixed modulus; and it is this analogy with equations which gives congruences (as such) one of their chief claims
to attention.
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