240 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
We want first any number with exponent 2. 
Suppose we take = 26 and a.j = 36 each with exponent 2 (for moduli 3® and 37 
respectively). 
Then we get 
= 703. 26 + 297. 36 (mod 999) 
= 998 (mod 999) with exp 2. 
We want next a number with exponent 4. 
Suppose we take 
and 
Then we get 
aj = I 
a, " 6 (with exp 4, mod 37). 
a= 703 + 297. 6 (mod 999) 
= 487 (mod 999) with exp 4. 
Lastl}^ we want two independent generators with exponent 3^. 
We first form unitary generators with exponents 3^ 
andi 
= 4 otj = 1 gives 112 with exp 3®, 
rt^ =: I = 12 gives 271 with exp 3'. 
We may take for our 3-powmr-exponent generators any two numbers of the form 
112 *‘ 271% 112''^ 27 U'^ provided that 
Suppose we take it 
and then tlie generators are 
•/ 
? o 
is prime to 3. 
1 1 
2 1 
112. 271 = 382 (mod 999) 
and 
]12b 271 = 826 (mod 999). 
Thus we have 
998 ■with exp 2, 
487 with exp 4, 
