FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 251 
2 + ^ 
3 + ii 
2 + 2i 
2 + 6 ^' 
7 + 5i 
Si 
1 + 7 ^ 
4 + 6^ 
2 + 7 i 
G + 7^■ 
5 4- 2 i 
8 
7 + 
6 + 5i 
7 + 7^ 
7 + 37 
2 + 4i 
7 
8 + 27 
5 
7 + 2t 
3 2t 
4 -f- 7 
1 
after which the set of twenty-four terms constantly repeats itself. 
(ii.) The proof is identical with 2, that if a has exponent t mod m andcP= 1, then 
t divides s. 
(hi.) The proof of Fermat’s theorem (mod m) is identical with that iii 
(3) if for “ the (/> {m) numbers less than m and prime to it,” we substitute “ the d) (m) 
residues of m that are prime to it.” 
Hence the corollary, that the exponent of any number, modulus m, is a divisor of 
d) (in). 
Example. —Mod 9 = 3'. d) (3~) = 3^^ — 3~ = 72. In Proposition (i.) we saw that 
the exponent of 2 -p 7, mod 9, is 24, a divisor of 72. 
(iv.) The proof is identical with that of (4). If a has exponent t, mod m, then cU 
has exponent r : where t — kt and k is the G.C.M. of s and t. 
2 K 2 
