238 MR. G. T. BENNETT ON THE RESIDUES OE POWERS OF NUMBERS 
The principal factors of the exponents of the generators are 
2. 2. 2. 3. 5. 
We form first the numbers with exponent 2 to generate the 2-exponent numbers. 
These numbers expressed in terms of the unitary generators, 155, 265, 197, are 
given by 
No. 
Ind. of 155. 
Ind. of 265. 
Ind. of 197. 
197. 
0 . 
0 . 
1 . 
265. 
0 . 
1 . 
0 . 
155. 
1 . 
0 . 
0 . 
111 . 
1 . 
1 . 
0 . 
43. 
1 . 
0 . 
1 . 
153. 
0 . 
1 . 
1 . 
307. 
1 . 
1 . 
1 . 
Of these (by Proposition 30) we can take any three such that the determinant 
formed by their indices is prime to 2, he., odd. 
For example, 
1 0 1 
0 1 1 = - I, 
1 1 1 
and so 43, 153, 307 may be taken as generators. 
Secondly we need a number with exponent 3. 
Of these there are two, 177 and 221, Let us take 177. 
Lastly we need a. number wdth exponent 5. 
Of these there are four, 113, 141, 169, 225. Let us take 113. 
We have now 43, 153, 307, independent generators with exponent 2, 
177 with exp 3, 
113 „ ,, 5. 
These we may combine in any manner we please as products, taking only one from 
each set in each product. 
Let us take 43. 177. 113 = 107 (mod 308) with exponent 30. 
We have thus obtained three generators 
107 with exp 30' 
153 „ „ 2 
307 „ „ 2 
which generate the complete set of ^(308) numbers. 
