128 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

HE<Li>k

Then, since I = k (mod x), by the proof just given
a' = a* (mod m)

Therfor a* = a' (mod m)

Ex.1. 3* =1 (mod 5) and 7 = 3 (mod 4).
Hence 37 = 3% (mod 5)

Ex.2. 4%=1 (mod 7) and 14 = 2 (mod 6).
Hence 4" = 42 (mod 7)

This theorem includes the preceding one as a particular case,
the case when I = 0.

I

305. Def. An integer a is said to be an even multiple of
an integer b, if a >> b and a : b is even; a is said to be an odd
multiple of b, if ¢ >> b and @ : b is odd.

For example, 4 is an even multiple of 2, 6 is an odd multiple
of 2; 6 is an even multiple of 3, 9 an odd multiple of 3.

306. Th. If an integer a is an even multiple of an integer b,
then a > 2b, whence a = 0 (mod 2b), and conversely.
If a is an even multiple of b,

a = ¢b, where ¢ is an even integer. § 305.
Since g is an even integer, ¢ = 2n, where 7 is some integer.
Hence a = (2n)b = n(2b)
Therfor a>2b

For the proof of the converse reverse the steps.

307. Th. If an integer a is an odd multiple of an integer b,
then a — b >> 2b, whence a = b (mod 2b), and conversely.
For the direct theorem we have

a = ¢b, where ¢ is odd. § 305.
Hence a — b= (¢ — 1)b, where ¢ — 1 is even.
Thus @ — b is an even multiple of b.
Therfor a — b > 2b § 306.
And a = b (mod 2b).

For the converse we reverse the steps.

308. Th. If a, m, and x are positiv inlegers,
a* = — 1 (mod m), and k is a positiv or zero integer; then,