FOR ANY COMPOSITE MODULES, REAL OR COMPLEX. 
213 
i.e., the multiplication of numbers expressed thus is simply ellected by forming the 
products of the coefficients of . . . 
Corollary (i.). If 
a = + . . . + (mod m), 
then 
cd = + a/+ • • • (mod m). 
Corollary (ii.). If 
a = + . . • (mod m), 
then a is congruent to the product of the numbers 
«! = + £3 + • • • (i^iod m), 
«o = ^1 + ^3^3 + . . . (mod m), 
&c. 
Or, since + . . . = 1 (mod m) {cf. Prop. 10), 
r«i = (a^ — l) + 1 (mod ri). 
< O3 = {ci.2 — 1) + 1 (mod m), 
Examples .—To rind a, modulus 308, so that 
a = OL^ (mod 4) 
= (mod 7) 
= ag (mod 11) 
7. 11, = 1 (mod 4) 4. 
3. 3. Xy = 1 (mod 4) 
= I (mod 4 ) 
^1 = 77 (mod 308) 
and so 
11. a’o 
= 1 (mod 7) 
= 1 (mod 7) 
= 4 (mod 7) 
= 176 (mod 308) 
4. 7. Xg = 1 (mod 11) 
G. Xg = 1 (mod 11) 
Xg = 2 (mod 11) 
l^g = 56 (mod 308) 
a = 77 + 176 + 56 ag (mod 308), 
a = 3 (mod 4) 
= 6 (mod 7) 
= 1 (mod 11) 
then 
a = 77.3 + 176. 6 + 56. 1 (mod 308) 
= 111 (mod. 308), 
