222 
MR. a. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
and 
€(, = (mod 7), 
a = ao^o + (mod 112). 
7xq=1 (mod 2^) 
2%^ = 1 (mod 7) 
Xq = 7 (mod 2^) 
25c^ = 1 (mod 7) 
= 49 (mod 112) 
Xi = 4: (mod 7) 
= 64 (mod 112), 
We will take 
a — 49a^ + 64aj (mod 112). 
and 
Then 
and 
/ = 7(=23-1) 
^0=3 (having exp 2^), 
= 3 (a primitive root of 7). 
(/ - 1) ^0 + 1 = 6. 49 + 1 = 295 = 71 (mod 112), 
(S-o - 1) fo + 1 = 2- ^9 + 1 = 99 (mod 112), 
(S'! — 1) fi + 1 = 2. 64 + 1 = 129 s 17 (mod 112); 
71 with exp 2 
99 with exp 4 |» generate the 48 numbers prime to 112. 
17 with exp 6 J 
The following table gives the indices corresponding to any number : — 
Numbers 1. 3. 5. 9. 11. 13. 16. 17. 19. 23. 25. 27. 29. 31. 33. 37. 39. 41. 43. 45. 47. 51. 53. 55. 57. 59. 
Ind. o£ 17 01 52430 1524 3 0 1 5 2 4 30 152 4 301 
Ind. of 99 011233 2 010 2 33 2 010 2 3 321 1 023 
Ind. of 71 0010011 0010 01 1 011 00 110 1 100 
Numbers 61. 65. 67. 69. 71. 73. 75. 79. 81. 83. 85. 87. 89. 93. 95. 97. 99. 101. 103. 107. 109. Ill 
Ind. of 17 52430152430152430 1 5 2 43 
Ind. of 99 30110232011023201 1 0 3 32 
Ind. of 71 1001100100 1 101100 1 1 0 11 
We proceed now to the point we have been approaching from the beginning, viz., 
the investigation of the mode of formation of and the relations among the most 
general set of numbers capable of generating the (f) [m) numbers (modulus m) which 
are j)rime to the modulus m. 
