CONGRUENCE 159

gruences, the processes of checking are themselves called cast-
ing out the nines (1), threes, elevens, etc., respectivly.

394. The methods of checking just described are good
methods, because they are so easily applied and because they
involve all the digits of the numbers operated upon.

395. Evidently we have in the sexidenal system methods of
casting out the threes, fives, and fifteens identical with those
for casting out the threes and nines in the decimal system;
and a method of casting out the seventeens identical with
that for casting out the elevens in the decimal system.

396. We have described how congruences may be used to
check arithmetical computations. If the check fails to work,
we may be sure there is an error either in the original com-
putation or in the work of applying the check. As the work
of checking is usually very simple, the error in such case is
most likely to be in the original work.

However, if the check works, we cannot therfor be sure
that the original work is correct. Even if the check has been
applied perfectly, there may be errors in the original work
which the check fails to disclose.

For example, an error like the following would not be
reveald by casting out the nines:

632

427

4424
1264
2528

42344

() The method of checking multiplication by casting out the nines is
very old. Ball says (l. c., p. 193) that it was invented by the Arabs. It
seems strange, if tru, that, as Lapparent remarks, people should not have
noticed the corresponding methods of checking other arithmetical processes.