In regards to repeating decimals and cyclic numbers: I understand that many of them, while multiplying by certain integers will produce a number with the same variant of digits, however my question is in regards to the patterns already occurring within some of them.
Many fractions that convert into a number with repeating decimals, will exhibit a pattern within the repetend. This pattern is made of a consecutive list of numbers growing by the same multiple.
example: $1/31=0.032258064516129...$., the repetend is $032258064516129$ which can be broken into the numbers $032258$, $064516$, $129032$ that are related by multiples of 2: $$032258∗2=064516, \quad 064516∗2=129032.$$
example: $1/13=0.076923....$, the repetend is $076923$ which can be broken into $0769$, $2307$ that are related by multiplying by 3: $$0769∗3=2307$$
example: $1/17=0.0588235294117647....$, the repetend is $0588235294117647$ which can be broken into the numbers $5882352941$, $11764705882$ that are related by multiplying by 2: $$5882352941∗2=11764705882$$
example: $1/19=0.052631578947368421...$, the repetend is $052631578947368421$ which can be broken into the numbers $05263$ , $15789$ , $47367$ , $142101$ (think of the initial 1 in 142101 being removed from the 8 in the preceding 47368) that are related by multiples of 3: $$05263∗3=15789, \quad 15789∗3=47367, \quad 47367∗3=142101$$
example: $1/21=0.047619...$, the repetend is $047619$ which can be broken into the numbers $0476$, $1904$ that are related by multiplying by 4: $$0476∗4=1904$$
What are these multiples? That is, given 1/$n$ with a repeating decimal, what multiple connects the different pieces of the repetend? (And how long are those pieces?)