first remainder, we divide 10 r by b , and, at the next step after the second remainder, we divide 10 b - 10 r by b . Now, since the sum of 10 r and 10 b - 10 r is divisible by b , the two remainders from these new steps must be such as added together will give b , and so on; and the quotients added together must give 9, for the sum of the remainders 10 r and 10 b - 10 r yields a quotient 10, of which the two remainders give 1.
If ¹/₅₉ and ¹/₆₁ be taken, the repeating parts will be found to contain 58 and 60 figures. Of these we write down only the first halves, as the reader may supply the rest by the complemental property just given.
01694915254237288135593220338, &c.
016393442622950819672131147540, &c.
Here, then, are two numbers, the first of which multiplied by any number under 59, and the second by any number under 61, can have the products formed by carrying certain of the figures from one end to the other.
But, b being still prime, it may happen that remainder 1 may occur before b - 1 figures are obtained; in which case, as shewn, the number of figures must be a measure of b - 1. For example, take ¹/₄₁.