132 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Using other moduli for the same example, the formula
a' = q't/ + 7' gives the following results.

252X2+3(mod5)
2=0X0+4+ 2 (mod 3)
5=3X0+4+5 (mod9)
5= 0 + 5 (modH6)
1= 0 + 1 (mod?2)
Involution can also be easily checkt by congruences.
For example, = 1 (mod 3)
Hence 43 = 13
Or 64 = 1, which is easily verified.

Evolution, when the root is exact, can be checkt by checking
the corresponding inverse operation of involution.

Thus, if V1728 = 12, then 1728 = 123,

Now 12 = 2 (mod 5)

Hence 122=23=8=3

Or 1728 = 3, which iseasily verified.

Logation, when the numbers involved are positiv integers,
can be checkt similarly.
For example, if logi; 1728 = 3, then 1728 = 123,

RADIX SYSTEMS AND CONGRUENCES

311. For numbers written in a radix notation, the usual
way, there are certain general rules of congruence that may
be stated in terms of the radix. We will work out some of
these rules and others deduced from them by supposing the
radix ten, the usual radix.

Since sixteen would be an ideal base, it will be interesting
also to interpret some of the general rules for the case when
the radix is sixteen.

We will suppose then that the integer a is written in the

form @n@n_1 * * + @100 (radix 7). See § 107.
Some of our results will be tru even if the digits ao, @i, as,
-«+, a, are not positiv or zero and numerically less than 7;

other results require that the digits conform to the usual rule
of being positiv or zero and less than 7.