1911 Encyclopædia Britannica/Number/Residues and congruences

27. Residues and congruences.&mdash;It will now be convenient to include in the term &ldquo;number&rdquo; both zero and negative integers. Two numbers $$a, b$$ are said to be congruent with respect to the modulus $$m$$, when $$(a-b)$$ is divisible by $$m$$. This is expressed by the notation $$a \equiv b \pmod{m}$$, which was invented by Gauss. The fundamental theorems relating to congruences are
 * If $$a \equiv b$$ and $$c \equiv d \pmod{m}$$, then $$a \pm c \equiv b \pm d$$, and $$ac \equiv bd$$.
 * If $$ha \equiv hb \pmod{m}$$ then $$a \equiv b \pmod{m/d}$$, where $$d = \operatorname{dv}(h, m)$$.

Thus the theory of congruences is very nearly, but not quite, similar to that of algebraic equations. With respect to a given modulus $$m$$ the scale of relative integers may be distributed into $$m$$ classes, any two elements of each class being congruent with respect to $$m$$. Among these will be $$\phi(m)$$ classes containing numbers prime to $$m$$. By taking any one number from each class we obtain a complete system of residues to the modulus $$m$$. Supposing (as we shall always do) that $$m$$ is positive, the numbers $$0, 1, 2, \dots (m-1)$$ form a system of least positive residues; according as m is odd or even, $$0, \pm 1, \pm 2, \dots \pm \tfrac{1}{2}(m-1)$$, or $$0, \pm 1, \pm 2, \dots \pm \tfrac{1}{2}(m-2), \tfrac{1}{2}m$$ form a system of absolutely least residues.