I‘J2 
MR. G. T. RIONXKTT ON THE RESIDUES OE POWERS OE NUMBERS 
r^et T Ije the ex})onent of cU 
e have 
therefore 
therefore 
A^'aiij, 
therefore 
therefore 
therefore 
therefore 
there] ore 
therefore 
= f (mod m), 
o" Er 1 (mod ui), 
= I (iiiod iil), 
(fC'^y = 1 (mod in), 
d' divides r. (Pro}). 2.) 
(-"U)^ = 1 (mod rii), 
= 1 (mod III), 
t divides /P, 
KT divides kctT. 
T dix ides (tT, 
7 divides 
1’ = 7. 
—Tlie ex})oiieiit of 3 for modulus 308 is 30 
suecessive powers are given iu the following table :— 
the residues of its 
Number . 
O 
0 
27 
I’owei' of 3 . 
1 
■) 
3 
ExjDonent 
30 
15 
10 
Number . 
47 
141 
115 
Power of 3 . 
11 
12 
13 
Exponent 
30 
5 
30 
Number . 
223 
53 
159 
Power of 3 . 
21 
22 
23 
Ex}ionent 
10 
15 
30 
81 
243 
113 
31 
93 
279 
221 
4 
5 
0 
7 
8 
9 
10 
1 5 
0 
5 
30 
15 
10 
3 
37 
111 
25 
75 
225 
59 
177 
14 
15 
10 
17 
18 
19 
20 
15 
2 
15 
30 
5 
30 
o 
O 
109 
199 
289 
251 
137 
103 
1 
24 
25 
20 
27 
28 
29 
30 
5 
0 
15 
10 
15 
30 
1 
where the ex})onents are all immediately deducible from the proposition. 
