196 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
therefore, 
therefore, 
Again, 
therefore, 
therefore, 
therefore, 
therefore. 
{aa'a" . . . )^ = 1 (mod m), 
T divides T. (Prop. 2.) 
{aa'a" . . .)' = 1 (mod iii), 
a’a'^a"'^ . . . = 1 (mod m), 
T = 0 (mod t) 
7 = 0 (mod t') >(Prop. 7), 
7 = 0 (mod t") 
T divides 7 , 
T = 7 . 
Corollary.—li the exjDonents t, t'^ t", be all of them powers of the same prime, and 
a, a, a", . . . are independent generators, then the exponent of the product aa'a!' . . . 
is equal to the greatest of the exponents t, t,' f . , . 
(9.) If the exponent of a for modulus m is t, and for modulus n is t', and if m and 
n are co-jDrime, then the exponent of a for modulus mn is the L.C.M. of t and t'. 
Let t" = tr = t'r where t and t' are co-prime, so that t" is the L.C.M. of t and t'. 
Let the exponent of a for modulus mn be T. 
Thus we have 
a^ = 1 (mod m), 
and, therefore, raising to the power 7 , 
a^" = 1 (mod m). 
Similarly, 
a^" = 1 (mod n), 
and, therefore (since m and n are co-prime). 
therefore 
Again, 
therefore 
therefore 
a^" = 1 (mod mn), 
T divides t". (Prop. 2.) 
=1 (mod mn), 
cP = 1 (mod m), 
t divides T. 
