Repeating decimals in converted fractions, why these increments?a) 131=0.032258064516129⋯ ., the repetend is 032258064516129 which can be broken into the numbers 032258, 064516, 129032 that are related by multiples of 2:b) 113=0.076923⋯. , the repetend is 076923 which can be broken into 0769, 2307 that are related by multiplying by 3:c) 117=0.0588235294117647⋯ , the repetend is 0588235294117647 which can be broken into the numbers 5882352941, 11764705882 that are related by multiplying by 2:d) 119=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:e) 121=0.047619⋯ , the repetend is 047619 which can be broken into the numbers 0476, 1904 that are related by multiplying by 4:

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Nathanial Carey · Accepted Answer

Step 1The first thing to think about is finding the repeating decimal by long division. Taking&amp;nbsp;117&amp;nbsp;for example, your first division starts with a remainder 1. You append a 0 to it and try to divide 17 into 10. It goes 0 times, which gives the leading 0 in the repeat. You then pass a remainder of 10 to the next digit. You pull down another 0, getting 100, divide 17 into it getting 5 with a remainder of 15. The 5 is the second digit of the repeat and the 15 goes on for the next digit.If you have a full length repeat like 17, you must use each of the remainders from 1 to&amp;nbsp;17−1&amp;nbsp;once. When you repeat a remainder you have finished the repeat. For 13, which has only a six digit repeat you only use six remainders. This explains all the relationships you show.After the&amp;nbsp;nth&amp;nbsp;digit the remainder is&amp;nbsp;10nmod17. We can find that&amp;nbsp;1010≡2±mod17&amp;nbsp;That means after 10 digits you pass a remainder of 2 to the next digit. The repeat from there will clearly be twice the repeat starting from the front, where the remainder is 1.

Repeating decimals in converted fractions, why these increments? a)

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