94 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

is a remainder got by dividing the predecessor of the preceding
number by the preceding number (or such that of each successiv
three numbers in this series the third is a remainder got by dividing
the first by the second, which is equivalent to saying that the first
is congruent to the third with respect to the second (§ 202)), then
a (X b is the numerical value of the divisor that gives the zero
remainder.

For uniformity’s sake let us set @ = 7_; and b = 7,.

Then our series is 7_1, 7o, 71, 72, ¥3, ***, ¥n_1, "a, in Which
7 =)

By hypothesis 7; is a remainder got by dividing 7;_s by 7;,
where 1 = & = n.

Hence 7;_2 = 7. (mod 751) § 202.
Therfor 7,_2 R re-1 = 71 R 7x §§ 196, 185, 188.
In this equality putting successivly £ =1, 2, 3, ---, n,
we get
raRro=rRn=rQRra=r2Qrs= -+ =r, 1R,
But7z, =0and 7,_.1 X0 = |7.] § 170.
Hence a ® b = |7n|
For example, let ¢ = 22 and b = — 16.
Dividing 22 by — 16 with quotient 3, remainder is 70
¢ RN 16 ‘6 70 6l ‘ __1 ‘é ‘o’ 54
‘il 70 ‘‘ 54 ‘ ‘“ 1 [ (3 16
“l 54 (X3 16 ‘h ‘e 4 e ‘il o 10
‘e 16 (X By 10 ‘ol ‘i s 2 ‘¢ ‘l = 4
‘e o 10 ‘il R 4 [ [ 2 ‘¢ [ =i 2
‘“° B 4 [ sl 2 ‘l ¢ 2 ‘l ‘’ 0

We find 22 (X (— 16) = 2

It should be noted that, with the exception of the zero
remainder, the remainder of each division becomes the divisor
of the next division, the former divisor becoming the new

dividend.

The work of division may be performd according to either
of the following schemes.