SOLUTION: What are the first $5$ digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{17}$ is written in base $16$? (Express your answer as a

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Question 1207585: What are the first $5$ digits after the decimal point (technically the hexadecimal point...) when the fraction $\frac{2}{17}$ is written in base $16$?
(Express your answer as a five digit hexadecimal number. You do not need to include the subscript to indicate the base.)
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Do some long division in base 16 ("2" base 16 divided by "11" base 16) and verify the result.

      -----------------
   11 ) 2. 0 0 0 0 ...


           1
      -----------------
   11 ) 2. 0 0 0 0 ...
        1  1
        -------------
           F 0


           1 E
      -----------------
   11 ) 2. 0 0 0 0 ...
        1  1
        -------------
           F 0
           E E
           ---------
             2 0

The remainder is the same number we started with, so the digits "1E" will repeat.

ANSWER: 0.1E1E1

Check to see that this hexadecimal is equal to 2/17:

0.1E1E1E... base 16 = ...

1%2F16%2B14%2F16%5E2%2B1%2F16%5E3%2B14%2F16%5E4+...

%281%2F16%2B14%2F16%5E2%29%2B%281%2F16%5E2%29%281%2F16%2B14%2F16%5E2%29+...

%2816%2F16%5E2%2B14%2F16%5E2%29%2B%281%2F256%29%2816%2F16%5E2%2B14%2F16%5E2%29+...

%2830%2F16%5E2%29%2B%281%2F256%29%2830%2F16%5E2%29+...

Use the formula for the sum of an infinite geometric sequence...