106 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

251. Th. If la + mb = 1, each of the integers I, a is prime
to each of the integers m, b.
Since la +mb = 1,la = 1 + —(mb)

But 1 4+ — (mb) 11 — (mbd) § 198.
Hence la 1| — (mb)
And la || mb § 159.

Therfor the conclusion of the theorem follows by § 633,
Elements of the Theory of Integers.

The theorem is also easily proved by showing from the
hypothesis that any common factor of either / or @ and either
m or b is a factor of 1.

252. Th. In the series of I's and m's given in § 239 each I,
except the last, is prime to its follower, each m, except the last,
to its follower, and each 1 to the corresponding m.

This follows from §§ 244, 251.

253. Th. Ifla + mb =a XD, thenl ]| m.
By the hypothesis I[a : (e R b)] + m[b: (e R )] =1
The theorem then follows from § 251.
This theorem may be stated: In whatever way a R b is ex-
prest in the form la + mb, | and m are prime to each other.
For example, 1 X 154 (— 1) X 12 =15 12 =3
and (—11) X 15 + 14 X 12 = 3
Here 1 ]| — 1 and — 11 ]| 14.

254. The theorem of § 239 can be put in another form as
follows:

Th. Having given two integers a and b, of which b —= 0, if
71 15 any of the numbers 71, To, 71, 72y 73, ** %4 T of § 226 and gy,
where 0 < k < n + 1, is the guotwnt used in dwzdmg 7r—2 by
7r—1, and we set

Sq1 = 1 t_1 — 0
So = 0 Ty = 1
Sk = Sp—2 + QrSk—1 bk = tr—2 + Qulr—1;

then for all values of k from — 1ton
= (= D*(swe — #:b),