IKDETEEMINATE EQUATIONS AND CONGEUENCES. 
321 
and 
M, 0 , 0 , 
. 0 , A., „ . 
..Ax 
0 , M, 0 , 
• ^ 5 Aj^ 1, . 
.. Aa 
0 , 0 , M, 
• ^ 5 -^3, 15 • 
.. Ag. 
0,0,0, 
IVl, A„^ 1, . 
• • A„ 
M, 0 , 0 , . 
0 5 A,,„ . 
•Ax. 
0 , M, 0 , . 
0 5 Aa, X, . 
.. Aa, 
0 , 0 , M, . 
0 5 Ag^ 1, . 
’ Ag. 
0,0,0,. 
M, A„_ 1, . 
.. A„, 
(83.) 
(84.) 
are to be equal to one another. Now the first of those greatest common divisors is 
e\idently the greatest common dmsor of 
M", M"-‘ Vi, Va, M v»-i, V» ; 
which, for brevity, we shall represent by the symbol 
[M”, M"-* Vi, M"-" V 2 , • • • V„-i M, v»] (85.) 
Let M=:P X Qx R • . P, Q, R, • • • denoting powers of different primes ; we may then, 
in (85.), replace M by P, Q, R, . . . successively, since 
[M", M"-' Vi, • • • M v„-n V»] 
=[P", P"-’ Vi, P v„-i, v«] X [Q", Q""' V, • • • vj X . . . 
If P di\ide any one of the numbers — ^ . . . . , let be the least of them that it divides ; 
^ V.-i V«-i 
also let P.: 
P, 
Vy 
Vi-I 
; so that Pi=P, if i>s. Then 
[P”, P”*-* Vn • • • • P Vn-l, Vn] 
p w El pn-l Vi pn-2 ^ V? 
-r.X|^p , r p , r p^, ...-p^ 
p ^Fpn-l pn-2 ^ pn-3 ^ YjL 
l_ Vi Vi Vi 
obser'ving that is prime to P [if s>l], and that we may therefore divide the last a 
numbers by ; and may then omit p^, which is divisible by P""’. Continuing this 
process, we find 
[pn, pn > . . . . Pv„_„ Vn] 
r 
= P,XP,X . . . xP,-i 
Vn 
pn-fi + 1 -pn-s pn-5-l^£±3 
'*■ r-7 ’'*• r7 
V«-i Vs-1 V«~i 
