FOR ARY COMPOSITE MODULUS, REAL OR COMPLEX. 
239 
Conversely, we can verify that 107. 153. 307 are complete generators. 
First we find that 
107 has exp 30 
and so the exponents have the necessary principal factors. 
107 = 107ih IO 710 . 107® (mod 308), 
= 43. 177. 113 (mod 308), 
where 
43 has exp 2' 
177 ,, „ 3 . 
113 „ „ 5., 
We only need to show that 43, 153, 307 are independent; and since the indices of 
these when referred to the unitary generators are 
43, 1 0 1 
153, 0 1 1 
307, 1 1 1 
and the determinant is prime to 2, we see that the generators 43. 153. 307 are 
independent. 
Let us form generators for modulus 999 —• 3®. 37. 
m = 3®. 37, 
^{m) = (3h 2)(3h 2^). 
The principal factors of the exponents of the generators are 
2 . 21 3 h 32. 
If 
a = ot-i (mod 3®), 
= (mod 37), 
« = (mod 3®. 37). 
37ajj = 1 (mod 3®) 
10a;, = 1 (mod 27) 
27 x 3 = 1 (mod 37) 
.^3 =11 (mod 37) 
^0 = 297 (mod 999) 
a;, =r 19 (mod 3®) 
^1 = 37. 19 = 703 (mod 999) 
and, therefore. 
a = 703ai + 297a3 (mod 999). 
