254 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Ill particular if the separate exponents are all jiowers of the same then the 
exponent is equal to the greatest of them. 
(ix.) Proof the same as for (9). 
If the exponent of a for modulus m is t, and for modulus n is i', and. m and n are 
co-prime, then the exponent of a for modulus m?i is the L.C.M. of t and t'. 
Also the corollary, if the exponents of a for moduli m, m, m”, . . . are respectively 
t, t', t",. . . m, m, m" . . . being co-prime, then the exponent of a for modulus mm'm"... 
is the L.C.M. of the numbers t, t', t'. . . 
Exam2ole. 
1 + f = 4 (mod 2 -b i), 
and the residues of its poivers are 4. 1. Hence the exponent of 1 + for modulus 
2 -f f is 2. 
1+1=6 (mod 3 + 2t), 
and the residues of its powers are 6. 10. 8. 9. 2. 12. 7. 3. 5. 4. 11. 1. Hence.the 
exponent of 1 + f for modulus 3 + '2i is 12. 
The moduli 2 + q 3 + 2i are co-prime, therefore the exponent of 1 + f for modulus 
(2 + ^) (3 + 2^■) is 12. 
1 + f = 19 (mod 4+7/), 
and the residues of its powers are 
19. 36. 34. 61. 54. 51. 59. 16. 44. 56. 24. 1. 
Exctmple. 
1 + 2i has exp 8, mod 3 ; 
the residues of its powers are 
1 + 2h i. 1 + i. 2. 2 + i 2i. 2 + 2h 1. 
I + 2f = 11 (mod 3 + 2f), and has exp 12 ; 
the residues of its powers are 
11. 4. 5. 3. 7. 12. 2. 9. 8. 10. 6. 1. 
1 + 2{ = 26 (mod 6 + 1), and has exp 3, 
the residues of its powers are 
26. 10. 1. 
Hence, since 3,3+ 2i, 6 + 2 are co-prime, 1 + '2i has for modulus 3 (3 + 2i) (6 + i) 
exponent = L.C.M. of 8, 12, 3, i.e., 1 + 2i has exponent 24 for modulus 48 + 452. 
The residues of its powmrs are 
