CONGRUENCE 111

Ex. 3. Ex. 4.

  

257. Th. If a is a positiv or zero integer and b a positiv
integer, these numbers, r_y and v, and the remainders ry, s, 73,

*, Tn, found by the algorithm of Euclid and corresponding to
exact or lower approximate quotients, can be exprest in one or
other of the forms sa — tb or — sa + tb, where s and t are positiv
or zero integers, all the r's with odd subscripts taking the form
sa — tb, the others the form — sa -+ tb.

258. Th. If 0 <k <n-+ 1, if sp and &, are as in § 254,
and dk = Sk_1tk e Sklfk_1, then dk = — dk—-l
and dr, = =+ 1, according as k is even or odd. §§ 243, 244.

259. Th. If a is a positiv or zero integer and b is a positiv
integer, two positiv or zero integers s and t can be found, such
that

sa —tbh = £ (a R 0d).

Fora(®b = 7,1 = =+ (sa — tb).
For example, 3 X 39 —5 X 24 =—-3;0X0—1X5=—375

260. Th. If a is a positiv or zero integer and b is a positiv
integer prime to a, two positiv or zero integers s and t can be
found, such that

sa —th =+ 1.
For example, 3 X 23 —4 X 17=1;0X0—-1X1=—1

() Altho the author workt out independently the form indicated in this
table, Moritz B. Cantor in his Vorlesungen iiber Geschichte der Mathematik,
Vol. 2, p. 696, mentions a similar diagram as having been used for com-
puting the convergents of a continued fraction by ‘‘Daniel Schwenter,
Geometria Practica von 1625 (III Auflage von 1641) S. 68 des zweiten
Traktates.”