CONGRUENCE 143

The theorem may also be easily proved by definition.
For example, 234 = 9 (mod r — 1)

347. Th. A number is congruent to the sum of its digits, if
the modulus is any factor of the immediate predecessor of the
radix.

348. Th. A number is or is not divisible by any factor of the
immediate predecessor of the radix according as the sum of its
digits is or is not divisible by that factor.

As special cases of §§ 347, 348 we have the following.

349. Th. In the decimal system a number is congruent to the
sum of its digits with respect to the modulus 3 or the modulus 9.
For example, 6038914 = 31 (mod 3 or 9)

350. Th. In the decimal system a number is divisible by 3
or by 9 if, and only if, the sum of its digits is so divisible.
For example, 504891 = 9, since 27 > 9
624981 —> 9, since 30 —>> 9.

351. Th. In the sexidenal system a number is congruent to
the sum of its digits with respect to either 3, 5, or G.

352. Th. In the sexidenal system a number is divisible by 3,
or by 5, or by G, if, and only if, the sum of its digits is so divisible.

353. Th. By means of § 347 a number whose digits are all
positiv or zero, or all negativ or zero, and which has more than
one digit may be reduced to a one-digit number with respect to
the modulus r — 1.

Thus, if the radix is ' and modulus 9,

273647 = 29 =11 = 2

989987979997 = 100 = 1
If the radix is 4 and modulus 3,

231032 =23 =11 = 2