Repeating Decimals, II

Abstract

In this chapter we examine the orders of elements of ℤ _t , for any number t, and complete our study of repeating decimals. These subjects are closely related, for we showed in Chapter I-12 that the base a expansion of u/t, ${u\over t}=(.a_1a_2\ldots a_da_1a_2\ldots a_da_1\ldots)_a\ =(.a_1a_2\ldots a_d)_a,$ is repeating:

$$ {u\over t}=(.a_1a_2\ldots a_da_1a_2\ldots a_da_1\ldots)_a\ =(.\overline{a_1a_2\ldots a_d})_a, $$

where d, the length of the period of repetition or, as we shall say for short, the period, is the smallest number n such that aⁿ ≡ 1 (mod t). That is, the period of u/t in base a is the order of a mod t.