CONGRUENCE 87

185. Th. a=b>Db=a

Fora —b>m>Db—a>m

On account of this theorem we may speak of two integers
being congruent to each other.

186. Th. a=b>D —a=—10>
Fora=bDa—03m>D —(a—b)3m>D—a—(—b)>m
D—a=—5»

I

187. Th. a = b (mod m) D a = b (mod — m)

188. Th. a=b~b=cDa=c
Fora —b3>m~b—c3>m>D@—0b+bB—c)>m
Da—c>m

189. Th. If in a series of integers a, b, ¢, * =, each, except

the last, is congruent to its follower, then the first 1s congruent
to the last, that is, a = 1.

190. Th. a = b (mod m) ~m > n D a = b (mod n)
Fora —b>3>m>nDa—0b>3n

191. Th. a =b (modm) Da=0> (mod m (X n)
This follows immediately from the preceding theorem.

102. Th. a=b(modm) ~b=c(modn)Da = c(modmQn)
This follows from the preceding theorem and § 188.

193. Th. a>>m~b>3>m>Da = b (modm)

In words: If both of two integers are divisible by the modulus,
they are congruent.

194. Th. a=b(modm) ~b>m>Da>=>m

In words: If two integers are congruent and either is divisible
by the modulus, so is the other.

Fora —b0b35>m~b>mDa>m

 

 

 

195. Th. a=b(modm) ~m>c~b>cDa>c
In words: If two integers are congruent and either is divisible

by a factor of the modulus, the other is also divisible by this
factor.