272 MR, G. T. BENNETT ON THE RESIDUES OE POWERS OF NUMBERS 
where 
aySy — 1 . 
For 
os-l 
«" — a, 
Therefore 
=13. 
and, therefore, 
(ahy^ ^ = a/3 = y [mod (1 + 
ah has exp 2 ^, and belongs to group y. 
Lemma (iv.). If 
and 
a has exp 2 *'^'" and belongs to group a, 
then 
h „ 2 ^ n „ 13, 
ah „ 2 *^" „ „ a. 
For 
oS + cr — 1 
ct = a, 
and, therefore. 
l/-^=J3, 
Therefore 
/ 7 \ 9 ^ + cr — 1 _ 
= a. 
Therefore 
ah has exp 2 ® and belongs to group a. 
Now, suppose that we take these generators with the necessary exponents. They 
must be chosen so that the numbers they generate are all incongruent. 
Let P 3 , P '3 be the generators. 
Then, g^, gh\ g^ E 
> 9 ^' [mocl (1 + must be impossible unless 
= ^y (mod exp g^), 
4 = 4' (iiiocl exp P' 3 ), 
i-i = H (mod exp g.^, 
i.e., g/-, g'J^= 1 [mod (1 + 7)^] must be possible only when 
Ji = 0 (mod exp g{}, = 0 (mod exp ^ 3 ), = 0 (mod exp pg). 
We shall now show that to render this so, the three o’enerators must belono' to 
three different groups, and that those three groups must not be any one of the seven 
sets of three, such as a, y, for u'hich a/3y = 1 . 
