92 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

223. In a radix notation a number is written in the form

At + Qua?™ P+ cos Far* 4 -+ ar + a

The numbers @,, an_1, *--, ar, ++-, a1, ao, called the digits,
are ordinarily supposed to be positiv or zero and numerically
less than » (§ 101). In what follows however we will fre-
quently find it convenient to allow these numbers to be negativ
or to be numerically equal to or greater than 7. In the proofs
there is no restriction on the values of the digits. See §§ 91, 92.

224. Th. If a number is written in a radix notation and the
modulus 1s added to or subtracted from any digit, the given number
will be congruent to the result.

For let a; be any digit.

Then ar = ar &= m (mod m) §§ 199, 200.
Hence A" + Q"L e Fapt* -0 Far + ay

=aut" F AQua?™ P+ -0 + (e Em)r*F + -0 F ar + ao
§§ 212, 207.

For example, 231 =
' 371 = 311 (mod 6)
263 = 203 (mod 6)
476 = 470 (mod 6)

If 7 is ten, 352 = 3T2 = 422 (mod 7)
493 = 4(17)3 = 573 (mod 8)
285 = 1(18)5 = 195 (mod 9)

We may of course add the modulus to or subtract it from
as many digits as we please.

Thus 2347 = 7897 (mod 5)

If 7 is ten, 3675 = 30T = 4230 (mod 5)

We may also use negativ digits.

Thus 2346 = 2542 (mod 8)
1256 = 4301 (mod 5)
3524 = 1920 (mod 4)

1321 = 4876 (mod 5)