FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
291 
r, = g.g- , . . c/;.‘ 
Us = g{^-^ g,y ^3’- . . . g;>^-^ 
= g{^^ g^^^ g^>^ . . . 
I 
i 
y (mod m) 
I 
J 
will be independent generators provided that the determinants foimed by the 
indices i, 
(i'll *22 • • • 4 a) 
are all prime to 
(.J 
The indices i which occur in any one of these determinants, are those which occur 
as indices of generators g all witli the same exponent; the generators U, in which they 
occur, having also this same exponent. 
The summary of Proposition (31) is also true of generators for complex moduli, 
(xxxii. and xxxiii.) With one modification. Propositions (22) and (23) hold good for 
complex numbers and primes. 
If in 
ax’‘ = 1) (mod in) 
a and m have G.C.M k, then 
a ^ / 1 «i\ 
- x" = - mod • 
K k\ K j 
'"a + 1 (1+1 
• 4ft) 
Each solution, x, of the second congruence gives N (k) solutions of the first, viz., all the 
numbers a; + 6’ —, where s is any one of the N [k) incongruent residues for modulus k. 
fC 
The solutions of the congruences which follow are intended as examples of these 
propositions and also as illustrations of the Tables placed in the Appendix. 
Example 1. 
7x^ = 3 (mod 4 + 2ft), 
From the tables 
(4) (2) 
3 = (3. 0) 
7 S ( 1 . 0). 
therefore 
( 2 . 0 ), 
.'(■ = (2. U) 
X = 9 (mod 4 + 2ft’). 
2 P 2 
therefore 
