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This is an investigation of infinitely repeating decimals such as
For convenience such sequences will be denoted as the repeating sequence underlined; e.g., 0.142857.
Only the numbers between 0 and 1 will be considered.
Let K be a sequence of decimal digits of length k considered as an integer. Then 0.K is an infinite sum
The value of the geometric series within the parentheses is 1/(1-10-k). Therefore
Thus any repeating decimal is a rational number. The form K/(10k−1) may not give the rational number in its simplest form. For example, for K=142857 both 142857 and 999999 are divisible by 142857 so 0.142857=1/7.
Let p and q be such that
where p is less than q.
Now consider which values of K are such that p=1. This requires that q be a factor of (10k-1). Take the case of k=6. The number 999,999 has a prime factorization of 3³7·11·13·37. Thus 1/3, 1/7, 1/11, 1/13 and 1/37 have such a repeating form. Note that 1/3=0..33333333… can be considered as a repeition of 333333 as well as a repetition of 3. But also 1/9, 1/21, 1/33, 1/39, 1/111 and so on will have that form, but likewise with 1/27, 1/45 and so forth.
Any multiple of (10k−1)/q for such values of q generated as a product of the prime factors of (10k−1) will be a repeating decimal such as
These multiples can only go up to (q-1).
Note that (10k−1) always has 9 as a factor. Therefore the crucial numbers for factorization are 111…111. The table below shows the factorization of integers to the base 10 which are sequences of 1.
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