156 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

We may also put the rule in the following form :

a = . WS 12 Y aF+6 Y a

k=0 (mod 6) k=1 (mod 6) k=2 (mod 6) k=3 (mod 6)
+ 4 Z a+ 5 Z ay,
k=4 (mod 6) k=5 (mod 6)

This form comes from the preceding, using the facts that
—1=6,—3=4,and — 2 =5 (mod 7).

The rule in this form may also be proved from §§ 102, 107,
since 10 =3, 102=2, 103=6, 10t =4, 10°= 5, 10° = 1
(mod 7).

391. Before applying the preceding rule it is usually best
to change the given number, if necessary, to a form in which
its digits are all positiv or zero, or all negativ or zero, and then
to apply the rule of § 382, which reduces the given number to
a number of not more than three figures.

‘For such a number the rule of § 390 becomes the following :

Th. 1In the decimal system, for a number of not more than
three figures, a = a4+ 3a, + 2a; (mod 7)

Thus we saw (§ 382) that 367,421,936 = 882 (mod 7)
By the present rule 882 =2 4+ 3 X 8 +2 X 8 = 42
=243X4=14=44+3X1=7

But 7 =0 (mod 7)

Hence 367,421,936 = 0 (mod 7)

Therfor 367,421,936 > 7

The present rule is bound to produce, as in this example,
by repeated application, a one-figure number, if the given
number is not such a number, but has all its digits positiv or
zero, or negativ or zero.

For |ao| + 10|as| + 100|az| > |ao| + 3|a1| + 2]as], if as
and a; are not both zero.

392. Th. In the decimal system, if the modulus is 13,

a= ( 2 =i ak> —3( o X ak)
k=0 (mod 6) k=3 (mod 6) k=1 (mod 6) k=4 (mod 6)

—4( Ry ak)
k=2 (mod 6) k=5 (mod 6)