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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.
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.
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