252 
MR. G-. T. BENNETT ON THE RESIDUES OF POWERS OE NUMBERS 
Example .—Taking again the successive powers of 2 + h 9, we find the 
exponent of each of the twenty-four numbers. 
No. 
7 = exp, 
2 + i 
1 
24 
3 -f 47 
2 
12 
2 4- 27 
3 
8 
2 + 67 
4 
6 
7 + 57 
5 
24 
8^ 
6 
4 
1 -f 77 
7 
24 
4 + 67 
8 
3 
2 + 77 
9 
8 
6 + 77 
10 
12 
5 + 27 
11 
24 
8 
12 
2 
7 + 87 
13 
24 
6 + 57 
14 
12 
7 + 77 
15 
8 
7 + 37 
16 
3 
2+47 
17 
24 
7 
18 
■ 4 
8 + 27 
19 
24 
5 + 37 
20 
6 
7 + 27 
21 
8 
3 + 27 
22 
12 
4 + 77 
23 
24 
1 
24 
1 
(v.) The proof is the same as that of (5). 
If the exponent of a is t, and of a is t\ and t and t' are co-prime, then the 
exponent of aa is tt'. 
And hence the corollary, that the same is true for any number of numbers : if 
a, a, a", . . . have exponents t, t', t", . . . co-prime, then aaa" . . . has exponent 
tt't" . . . 
Example .—The exponent of 3 for mod 1 -j- bi is 3. 
Its successive powers have residues 3. 9. 1. 
The exponent of 5 for mod 1 d- bi is 4. 
Its successive powers have residues 5. 25. 21. 1. 
Hence the exponent of 5 X 3 = 15 is 4 X 3 = 12. 
The successive powers of 15 have residues 15. 17. 21. 3. 19. 25. 11. 9. 5. 23. 7. I. 
