FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
193 
(5.) The exponent of a is t and of a is t' and t and t' are co-prime; then tiie 
exponent of aa is tt'. 
Let T be the exponent of aa. 
Then, 
a* = 1 (mod m) a'^' = 1 (mod m), 
therefore 
a*^' = 1 (mod m) a'"' = 1 (mod m), 
therefore 
{aaY' = 1 (mod m), 
therefore 
T divides tt'. (Prop. 2.) 
Again, 
{aa!Y = 1 (mod m), 
{aa'Y^ = 1 (mod m), 
= 1 (mod m), 
therefore 
t' divides Tt (Prop. 2), 
therefore 
t' divides T. 
Similarly, 
t divides T, 
and therefore (since t and t' are co-prime), 
tt' divides T, 
therefore 
T = tt'. 
Corollary. —It follows that if the numbers a, a, a", . . . have exponents (for 
modulus m) t, t' t" , . . . these exponents being all co-prime, then the exponent of 
aaa . . . i^ttt ... 
Example. —The exponent of 
23 mod 308 is 6 ; (23. 221. 155. 177. 67. 1), 
and the exponent of 
113 mod 308 is 5 ; (113. 141. 225. 169. 1). 
Since 5 and 6 are co-prime, it follows that the exponent of 
135 = 23. 113 is 30. 
Corollary. Examp)le. —The exponent of 
MDCCCXCIII.—A. 2 c 
