According to Midy’s theorem, if the period of a repeating decimal for , where p is prime and is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. For example 1/7 = 0.14285714285.. is a repeating decimal with 142857 being repeated. Now, according to the theorem, it has even number of repeating digits i.e. 142857. Further if we divide this into two halves, we get 142 and 857. Thus, on adding these two, we get 999 which is string of 9s and matches our theorem.
In Extended Midy’s theorem if we divide the repeating portion of a/p into m digits, then their sum is a multiple of 10m -1.
Suppose a = 1 and p = 17,
a/p = 1/17 = 0.0588235294117647…
So, 0588235294117647 is the repeating portion of the decimal expansion of 1/17. Its repeating portion has 16 digits and it can be divided into m digits where m can be 2, 4, 8.
If we consider m = 4 then 0588235294117647 can be divided into 16/4 = 4 numbers and if we add these 4 numbers then result should be a multiple of 104 – 1 = 9999 i.e,
0588 + 2352 + 9411 + 7647 = 19998 = 2*9999
Repeating decimal = 0588235294117647 588 2352 9411 7647 Extended Midy's theorem holds!
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