116 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

279. Th. a=b(modm) ~a>>c—~b>c>D
a:c=bz:c(mod (mXc):c);
and conversely, a>>c~b>>c—~a:c=b:c(mod(m Xc):c)
Da = b (mod m)
The direct theorem follows immediately from § 278 and the
direct part of § 218.
The converse follows from the converse part of § 218 and

§ 190.
This theorem includes § 218 as a particular case.
For,if m >>¢c,m X ¢ = |m]|.

Example. 54 = 6 (mod 8), 54 > 6,6 > 6
54 :6=06:6 (mod (8 X 6) :6)

280. Th. a=0b0(modm) ~a>c~b>c>D
a:c=b:c(modm:(mc)); and conversely.

This follows from §§ 187, 271, 279.

Form X c¢: |¢c| = |m| : (m R ).

281. Th. a=b(modm) ~a>>c—~b>c—~mllcD
a:c=b:c(modm);
and conversely, a>>c ~b>>c~m]lc—~a:c=b:c(modm)
Da = b (modm)
This follows from § 279, since, if m |l ¢, m X ¢ = |mc|
= |m||c|, or from § 280, since, if m 1l ¢, m Q¢ = 1.

Setmod 5), 30 33, 153% 3,511 3,
15 : 3 (mod 5)

Example. 30
and 30.:3

282. From § 216 it follows that we may always multiply the
members and the modulus of a congruence by any non zero
integer; from § 218 that we may always divide the members
and the modulus by any integer by which they are divisible.

From § 212 it follows that we may multiply the members
by any integer without altering the modulus; from § 217 that
we may divide the modulus by any integer by which it is
divisible, leaving the members unchanged.

But if we multiply the modulus by any non zero integer, in
order to be sure that the congruence holds we may, tho not

I