FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
247 
and 
therefore 
4 + = (2 + i) (3 + 2i) 
60 -h 105i = 3 (2 - {) (2 + if (3 + 22). 
The number of incongruent residues for any modulus m. 
The number of incongruent residues (including zero) is the norm of the modulus. 
Thus if w = a + ^i> the number of incongruent residues modulus m is + /3~. When 
the modulus is a pure number, m = p, then the number of incongruent residues is p^. 
Set of numbers all incongruent for modidus m. 
Let m — d {a /Si) where a and /3 are co-prime. 
Then the N (m) = d^ {od + /3^ ) numbers x -|- iy, where x has any one of the values 
0, 1, 2, . . . c/ {od + y8^) — 1, and y any one of the values 0, 1, 2, 3, ... — 1, are 
a complete set of N (m) incongruent residues for mod m. 
We have two special cases :— 
(i.) If wi = a + jii, the residues are 0, 1, 2, . . . a~ + /3“ —1. 
(ii. ) If m = d, the residues are the numbers x iy, where x and y have each any one 
of the values 0, 1, 2, ... c/ — 1. 
For most purposes the above set of residues are most convenient. It is sometimes, 
however, a saving of arithmetical labour (in examples) to make use of the “ absolutely 
least residues,” i.e., the set of residues whose norms are not greater than half the norm 
of the modulus. We may also make use of “least positive residues,” i.e., the set of 
numbers whose norms are as small as possible consistently with each number having 
both its parts positive. 
In what follows we shall always use the above set of residues and we shall need an 
expeditious method for finding to which of them any given number is congruent. 
To reduce a number to its residue. 
Let X -b W be the number whose residue we wish to find for mod cZ (a -j- ^i), where 
a and /3 are co-prime. 
We have to find x and y so that 
where 
X -j-ZY = x iy (mod cZ a + t^i), 
X < d {ad + /Q^) 
y <d 
>and both are positive numbers. 
Let 
X + iY — (.T -b iy) + (^ + iy) (« + f^i) d. 
