194 
:\[R. G. T. BENNETT ON THE RESIDUES OP POWERS OF NU:\IBERS 
M 1 is 2 ' 
221 is 3 >therefore the exponent of 111. 221, 113 — 3 is 30. 
113 is 5 
J 
(6.) Let a have exponent t and a' exponent t' and suppose that t and t' are not 
co-prime. Then, if t and t' contain no prime factor raised to the same power in both, 
the exponent of aa^ is the L.C.M. of t and t'. 
Let 
t = KT 
»^ _ ‘ 
t — KT 
then 
• where r and t are co-prime, 
therefore. 
rO has exponent r (Prop. 4), 
cf" has exponent t (Prop, 4), 
{aa'Y has exponent tt (Prop. 5). 
So if aa has exponent T, then 
rn / 
J TT 
K C 
where tt divides T, 2 divides k and s is prime to tt. (Prop. 4.) 
Now since t and t' contain no prime raised to the same power in each, therefore tt' 
contains every prime factor which occurs in t and t' and therefore contains every 
prime factor which occurs in k. Hence 2 cannot divide k and be prime to tt' unless 
it be unity. 
Therefore, 
T = KTT = L.C.M. of t and t'. 
(7.) If a has exponent t, a has exponent t', a" — t", &c., for modulus on, and if, of 
the tt'f” . . . numbers cCd'^'a''" . . . (mod 7n) formed by giving to o' all values modulus t, 
to r' all values modulus t' . . no two are congruent ; and if 
afd^'d'^" . . . = 1 (mod ooi) ; 
then we must have 
cU = 1, fid = 1, a”^" = 1. . . . (mod m). 
For suppose that at least one of these congruences is violated. 
Say 
^ 1 (mod «?), 
and, therefore. 
.V ^ 0 (mod t). 
