PROP. 8. In the last case, _a_ᵇ⁻¹ divided by _b_ leaves a remainder 1. From the last, _a_ᵇ-_a_ leaves the same remainder as (_a_-1)ᵇ + 1-_a_ or (_a_-1)ᵇ- (_a_-1); that is, the remainder of _a_ᵇ-_a_ is not altered if _a_ be reduced by a unit. By the same rule, it may be reduced another unit, and so on, still without any alteration of the remainder. At last it becomes 1ᵇ-1, or 0, the remainder of which is 0. Accordingly, _a_ᵇ-_a_, which is _a_(_a_ᵇ⁻¹- 1), is divisible by _b_; and since _b_ is prime to _a_, it must (Prop. 2) divide _a_ᵇ⁻¹-1; that is, _a_ᵇ⁻¹, divided by _b_, leaves a remainder 1, if _b_ be a prime number and _a_ be not divisible by _b_. From the above it appears (Prop. 5 and 7), that if _a_ be prime to _b_, the set 1, _a_, _a_², _a_³, &c. successively divided by _b_, give a set of remainders beginning with 1, and in which 1 occurs again at _a_ᵇ⁻¹, if not before, and at _a_ᵇ⁻¹ certainly (whether before or not), if _b_ be a prime number. From the point at which 1 occurs, the cycle of remainders recommences, and 1 is always the beginning of a cycle. If, then, _a_ᵐ be the first power which gives 1 for remainder, _m_ must either be _b_-1, or a measure of it, _when b is a prime number_. But if we divide the terms of the series _m_, _ma_, _ma_², _ma_³, &c. by _b_, _m_ being less than _b_, we have cycles of remainders beginning with _m_. If 1, _r_, _s_, _t_, &c. be the first set of remainders, then the second set is the set of remainders arising from _m_, _mr_, _ms_, _mt_, &c. If 1 never occur in the first set before _a_ᵇ⁻¹ (except at the beginning), then all the numbers under _b_-1 inclusive are found among the set 1, _r_, _s_, _t_, &c.; and if _m_ be prime to _b_ (Prop. 4), all the same numbers are found, in a different order, among the remainders of _m_, _mr_, &c. But should it happen that the set 1, _r_, _s_, _t_, &c. is not complete, then _m_, _mr_, _ms_, &c. may give a different set of remainders. All these last theorems are constantly verified in the process for reducing a fraction to a decimal fraction. If _m_ be prime to _b_, or the fraction _m_/_b_ in its lowest terms, the process involves the successive division of _m_, _m_ × 10, _m_ × 10², &c. by _b_. This process can never come to an end unless some power of 10, say 10ⁿ, is divisible by _b_; which cannot be, if _b_ contain any prime factors except 2 and 5. In every other case the quotient repeats itself, the repeating part sometimes commencing from the first figure, sometimes from a later figure. Thus, ¹/₇ yields ·142857142857, &c., but ¹/₁₄ gives ·07(142857)(142857), &c., and ¹/₂₈ gives ·03(571428)(571428), &c. In _m_/_b_, the quotient always repeats from the very beginning whenever _b_ is a prime number and _m_ is less than _b_; and the number of figures in the repeating part is then always _b_-1, or a measure of it. That it must be so, appears from the above propositions.