FOK AlsY COMPOSITE MODULUS, REAL OR COMPLEX. 
253 
Example .—The exponent of 2, mod 7, is 3. The residues are 2. 4. 1. 
The exponent of 5 + 2f is 8. The residues are 5 + 2^, 6t, 2 + 2i, G, 2 + 5u i 
5 -f bi, 1. 
Hence the exponent of 10 + 4?! = 3 + is 3X8 = 24. 
The successive residues are 
3 + 4? 
3i 
2 + 2i 
5 
1 + 6i 
3 + 3i 
4 
5 + 2i 
5i 
1 + i 
4 + 3i 
4i 
5 + 5 1 
o 
6 + i 
4 + 4? 
3 
2 + 5? 
2/ 
6 + 6^’ 
(vi.) The proof is the same as for (6). 
Suppose that a has exponent t, and a has exponent t', and 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'. 
Example .—The exponent of 35 for mod 9 6i is 6. 
The residues of its powers are 35. 16. 14. 22. 29. 1. 
The exponent of 3 + 7i for the same modulus is 4. 
The residues of its powers are 3+7?’. 14. 15 + 2?. 1. 
Hence the exponent of 35 (3 + 7i) is the L.C.M. of 6 and 4 = 12. 
35 (3 + 7i) = 105 + 245?= 33 + 2? (mod 9 + 6?). 
The residues of its powers are 
33 + 2? 29 33 + i 22 2? 14. 
12+? 16 15 + 2i 35 18 +? 1. 
(vii.) The proof is the same as for (7). 
If a has exponent t, a has exponent t', a' has exponent t", &c. for modulus m, and 
if of the tt't" . . . numbers . . . (modulus m), formed by giving to r all values 
modulus t, to r' all values modulus t', &c., no two are congruent; then if 
we must have 
. . . = 1 (mod m), 
a-^ = 1 , a^' = 1 , . . . 
in other words, if a product of powers of numbers that are independent generators 
be congruent to unity, then each of these powers is itself congruent to unity. 
(viii.) Proof identical with (8). 
The exponent of the product of a number of independent generators is the L.C.M. 
of their exponents. 
