What is the maximum number of digits possible in the repeating block when you divide out the rational 1/46229. How do you know this?

Question

is it one less than the denominator i.e. 46229? Explain why

Answer 1

The maximum number of digits possible is always one less than the denominator. In this case the maximum number of digits possible in the repetend is 46,229-1=46,228. The reason is that given any number the number of possible remainders is always one less than the denominator or the divisor.

For example, express 4/7 as a repeating decimal. The possible remainders are 1,2,3,4,5, and 6 when you divide by 7.

You can't have 7 as a remainder because if you did then you divided incorrectly. If you get 0 as a remainder you have a terminating decimal. You don't always get the maximum number of digits in the repetend when you divide. You do however with 7 as a denominator.

 

   0. 5 7 1 4 2 8...

  ________________

7l 4.0 0 0 0 0 0

   0

 ___

   40

   35

 ____

     50

     49

 _____

        10

          7

  _______

           30

           28

    _______

             20

             14

    _________

               60

               56

      _____________  

                  4           (now the remainders start to repeat and therefore the digits in the repetend start to repeat)

another denominator that gives you the maximum number of digits in the repetend is 19, try one out and you will get a repetend 18 digits in length