What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$ ? Perform the division to check your answer.

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To do:

We have to find the maximum number of digits in the repeating block of digits in the decimal expansion of $\frac{1}{17}$.
Solution:

Dividing 1 by 17 using the long division method, we get,

17)100(0.0588235294117647

85

---------

150

136

---------

140

136

-----------

40

34

-----------

60

51

-------------

90

85

--------------

50

34

--------------

160

153

---------------

70

68

------------

20

17

------------

30

17

----------

130

119

-----------

110

102

-------------

80

68

-------------

120

119

-------------

1

This implies,

$\frac{1}{17}=0.0588235294117647............$

$=0.\overline{0588235294117647}$

The maximum number of digits in the repeating block of digits in the decimal expansion of $\frac{1}{17}$ is 16.

Updated on 10-Oct-2022 13:38:47