FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
197 
Similarly, 
therefore 
Therefore 
t' divides T, 
t" divides T. 
T = f. 
Corollary .—If the exponents of a for moduli m, m, m", . . . are respectively 
t, t', t ",... and the moduli are co-prime, then the exponent of a for modulus mmm" . . 
is the L.C.M. of t, t', t" . . . 
Exam'ples .—The exponent of 
3 for mod 4 is 2 "j 
3 for mod 7 is 6 •; 
3 for mod 11 is 5 J 
(3. 1.) 
(3. 2. G. 4. 5. 1.) 
(3. 9. 5. 4. 1.) 
Therefore the exponent of 
3 for mod 4. 7. 11 
The exponent of 
5 for mod 4 is 1. 
5 for mod 7 is 6 
5 for mod 11 is 5 
308 is 30. 
(5. 4. 6. 2. 3. 1.) 
(5. 3. 4. 9. 1.) 
and, therefore, the exponent of 
5 for mod 308 is 30. 
The exponent of 
9 for mod 4 is 1. 
9 for mod 7 is 3 (2. 4. 1.) 
9 for mod 11 is 5 (9. 4. 3. 5. 1.) 
and, therefore, the exponent of 
9. mod 308 is 15. 
(10.) If the exponent of a is t, and t = pqr . . . where p, q, r are co-prime factors 
of t, to express a as a product of numbers whose exponents are qj, q, r . . . 
Let 
P = 0 (mod qr . . .)] Q = 0 (modp^’ . . .) 
= 1 (modp) J =1 (mod q) 
These congruences determine one value each and one only (mod t) for P, Q, . . . 
