255 
FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
1 4 -2^’. 90 + ^. 1339 +L 692. 275 + l 1419 + 2l 1370 4-27 
1231. 1012 4- 2i. 291 4- i. 1015 4- i. 482. 
32 4-7 540 4- 27 827 4 - 27 211 . 244 4 - 27 969 4-7 439 4-7 
269. 1043 4- 7 741 4- 27 503 4 - 27 1 . 
(x.) Proof identical with ( 10 ). 
If the exponent of a is t, and t = pqv. . . where p, q, r . . . are co-prime factors 
of t, then a can be expressed as a product of numbers whose exponents are p, q, r. . . 
Example .—2 4 - i has exponent 24 for modulus 9. (See example Proposition iv.) 
Hence 
where 
and 
24 = 3.8 16 = 1 (mod 3)] 9 = 1 (mod 8). 
= 0 (mod 8) J =0 (mod 3). 
2 -j- 2 = (2 4- (2 -j- (mod 9), 
= ( 74 - 34 (2 4- 7i) (mod 9), 
7 4 - 3^ has exp 3 
2 -p 7i has exp 8 
(See example Proposition iv.) 
Example .—33 4- ’2i has exponent 12 for modulus 9 -f 67 (See example Pro¬ 
position vi.) 
12 = 3. 4 4 = 1 (mod 3) 9 = 1 (mod 4) 
= 0 (mod 4) =0 (mod 3). 
Therefore 
33 4- 2t = (33 4- 2^^ (33 4- (mod 9 4- 67) 
= 22 ( 154 - 27) (mod 9 4- 67), 
where 22 has exponent 3, and 15 4- 27 has exponent 4. 
(xi.) The number of numbers with exponent t, when the modulus is a prime (pure 
or mixed), is {t). 
Any exponent t is a divisor of <E> {p), p being the prime modulus (Corollary, Pro¬ 
position iii.). Exactly as in ( 11 ) we see that if there be one number with exponent t 
there are (j) {t) and no more. 
Now C, C, . . . being all the divisors of any real number d> (p), 
^ (h) + ^ (C) + • • • = {qj). 
Corresponding to each value t there aie numbers, or none with t as exponent. 
