236 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OP NUMBERS 
Tlie lemma proved iu tlie course of the last proposition (that if > 1^, then is 
divisible by p), shows that every element i, which is found to the left and below the 
determinants [aa), (bh), [cc) ... is divisible by i.e., every element in 
{ha) 
{ca) 
{ch) 
{da) 
{dh) 
{dc) 
is divisible by &c.. 
'The complete determinant [in, . . . J can be expressed as the sum of products of 
pairs of complementary determinants, one of these in each such product being the 
determinant formed by any a rows taken from {aa), {ha), (ca) . . , 
Now, any row taken from (ha), (ca), . . . has every element divisible by j:). Hence, 
the only determinant formed by a rows taken from (aa), (ha), (ca), . . . which is not 
immediately divisible by p, is the determinant {aa) = . . . v). 
Hence 
{in, • • • v) = (hn • • • ia) (4+i.a+i • • • (modp). 
Therefore, if (fn • • • i^^) is prime to p, then . . . 4a) and (4+i.a+u • • • v) are 
both prime to p (and vice versa). 
Again, take the determinant (4 + i.a+i> • • • v)j 
{hh) 
{he) 
{ch) 
(cc) 
&c. 
&c. 
This can be expressed as a sum of products of determinants in the same way : one 
determinant in each product being formed by (6 — a) rows taken from ihh), {ch) . . . 
Of these, only {hh) is not immediately divisible by p. 
