Question

What can the maximum number of digits be in the repeating block of digits in the decimal expansion of `1/17`? Perform the division to check your answer.

Answer 1

In `1/17`, the divisor is 17.

Since the number of entries in the repeating block of digits is less than the divisor, then the maximum number of digits in the repeating block is 16.

Dividing 1 by 17, we have

       0.0588235294117647...
`17)overline1.0000000000000000`
      -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

The remainder 1 is the same digit from which we started the division.

∴ `1/17` = `overline0.0588235294117647`

Thus, there are 16 digits in the repeating block in the decimal expansion of `1/17`.

Hence, our answer is verified.