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Residue

The word residue is used in a number of different contexts in mathematics. Two of the most common uses are the complex residue of a pole, and the remainder of a congruence.

The number b in the congruence a=b (mod m) is called the residue of a (mod m). The residue of large numbers can be computed quickly using congruences. For example, to find 37^(13) (mod 17), note that

37=3
(1)
37^2=3^2=9=-8
(2)
37^4=81=-4
(3)
37^8=16=-1,
(4)

so

37^(13)=37^(1+4+8)=3(-4)(-1)=12 (mod 17).
(5)

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