CONGRUENCE . 93

CASTING OUT THE MODULUS

225. When either of the above indicated operations gives
zero for the result in any place, we may indicate this by
crossing out, or cancelling, the corresponding digit.

Thus 234341 = 230301 (mod 4)

569434 = 569030 (mod 4)

If the result is not zero, we may similarly cross out the
original digit and put the new digit above or below.

Thus 734634 = 310230 (mod 4)
3% 2

If 7 is ten, 439628 = 351620 (mod 8)
351

Def. When the original digits are all positiv or zero and
the modulus is subtracted, this process may be called casting
out the modulus.

Other examples:

£97809 = 17001 (mod 8), casting out the 8’s,
§oi=¢

396957 = 306057 (mod 9), casting out the 9’s.

There are however more elaborate processes, to be described
later, of which the above process is a part, which also are called
casting out the nines, casting out the elevens, etc.

GREATEST COMMON FACTOR

226. Th. Having given two integers a and b, of which b —= 0,
if we determin a finite series of numbers

71, 72, 73, * = ° I'n;

of which only the last is zero and such that, starting with r1, each
number in the series
a, b1 71, 72, F3y o vty ¥n