198 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
has exp p (Prop. 4), 
&c. 
From the above congruences we obtain 
therefore, by addition, 
Similarly, 
P = 1 (mod^j), 
Q = 0 (mod p )), 
R, = 0 (mod 79 ), 
&c. 
P + Q + R + • • • = 1 (mod pi). 
P + Q + R + . . . = 1 (mod q), 
&c.. 
and, therefore, since p, (p r, . . . are co-prime, 
P “h Q “p R “h ... — 1 (mod t'), 
therefore 
a^\ a^. ...= a (mod m). 
And so a is expressible (in one way only) as the jDroduct of numbers cP, 
whose exponents are p, q . . . 
Example .—3 has exponent 30 mod 308. 
To express it as a product of 3 numbers with exponents 2 . 3. 5, 
therefore 
P = 0 (mod 15) 
= 1 (mod 2 ) 
P = 15 
Q = 0 (mod 10) 
= 1 (mod 3) 
Q = 10 
R = 0 (mod 6 ) 
= 1 (mod 5), 
R = 0, 
and, therefore, 
3 = 3^^ 31 *^. 30 = 111. 221. 113 (mod 308) 
where 
111 
221 
113 
has exp 2 
)j j) 3 >- 
s 
3? 3) 
Similarly 79 has exponent 30 (mod 308), and 
79 = 79^0 79*^ = 43. 177. 14 L (mod 30S) 
