1911 Encyclopædia Britannica/Number/Residues and congruences
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27. Residues and congruences.—It will now be convenient to include in the term “number” both zero and negative integers. Two numbers are said to be congruent with respect to the modulus , when is divisible by . This is expressed by the notation , which was invented by Gauss. The fundamental theorems relating to congruences are
- If and , then , and .
- If then , where .
Thus the theory of congruences is very nearly, but not quite, similar to that of algebraic equations. With respect to a given modulus the scale of relative integers may be distributed into classes, any two elements of each class being congruent with respect to . Among these will be classes containing numbers prime to . By taking any one number from each class we obtain a complete system of residues to the modulus . Supposing (as we shall always do) that is positive, the numbers form a system of least positive residues; according as m is odd or even, , or form a system of absolutely least residues.