90 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

This can be proved either by definition or from the pre-
ceding two theorems and § 188.

207. Th. The sum of the left members of any number of con-
gruences is congruent to the sum of the right members.
For example,a =b~c=d—~e=fDa+c+e=b+d+f

208. Th. (a=b) =(@—c=b—y)
209. Th. (a=b)=(c—a=c—0)
210. Th. ea=b~c=d>Da—c=b—4d

211. Th. Any algebraic sum of the left members of any num-
ber of congruences is congruent to the like algebraic sum of the

right members.
Forexample,a=b~c=d~e=fDa—c+e=b—d+ f

212. Th. a=bDac=bc
Fora—b>3>m>Dac—bc=(a—b)c>m>Dac—bc>m

213. Th. a=b>Dca=cb
214. Th. a=b~c=d>Dac=0bd

215. Th. The product of the left members of any number of
congruences is congruent to the product of the right members.
For example, a = b ~c =d —~ e = f D ace = bdf

216. Th. [a = b (mod m) ~ c—= 0] = [ac = bc (mod mc) ]

217. Th. a=b(modm) ~m>cDa=>b(modm : c)
This follows from § 190, since, if m > ¢ and m —= 0, then
0oL

218. Th. [a=b(modm) ~a>>c~b>c—~m>c
Da:c=b:c(modm:c);
and conversely a>>c—~b>>c—~m>>c—~a:c=b:c(modm:c)
Da = b (mod m)
For the direct theorem we have a = d¢, b = ec, m = nc.
Hence dc = ec (mod nc)
Therfor d = e (mod n) § 216.