118 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

(§ 206), whence, since km + In = m (Q n, we get the desired
result.
For the converse, since a = (kmc + Inbd) : (m R n)(mod X n),
a = (kmc +1Inb) : (m X n)(modm)
§ 190.
Since b = ¢ (mod m (R 1), ¢ = b (mod m (X 1)
Hence mc :(mQQn) = mb : (m(XQn) (mod (mQn) (m : m@n)))

Hence kmc : (m Qn) = kmb : (m (X n) (mod m) § 212
To the two members of this congruence adding nb : (m R n),
(kme 4 Inb) : (m (R n) = (km + In)b : (m R n)(mod m) § 204.

Therfor a = b (mod m)
Similarly ¢ = ¢ (mod 7) ()

For example, taking m = 4,7 = 6,a = 29, we find mXQQn = 2,
Xl e -1 l=1,0=1¢=
29 =1 (mod 4) 29 = — 1 (mod 6)

29 = [4(— 1)(— 1) +6(1)(1)] : 2 (mod 12)
29 = 5 (mod 12)
The theorem of § 276 is a special case of this theorem, got
by putting ¢ = b.
284. Th. Ifa = b (mod m), a = ¢ (mod n), m || n, and k
and l are two integers such that km + In = 1, then

a = kmc + Inb (mod mn);

conversely, if km + In = 1 and a = kmc + Inb (mod mn), then
a = b (mod m) and a = ¢ (mod n).

This theorem follows from the preceding theorem by §§ 171,
275, 187.

Example. Takem =3,n=4,a=7. Wefind 2t = —
$imd b ] e 3

= 1 (mod 3) 7 = 3 (mod 4)
7= — 5 (mod 12)
() The direct theorem discoverd Aug. 8, 1931, the converse and these
proofs the next day.