Question

What is the decimal representation of `17/5210` ?

Answer 1

`17-:5210=.003262955854`

Explanation:

Using a calculator, `17-:5210=.003262955854`.

Answer 2

`17/5210 = 0.0bar(0326295585412667946257197696737044145873320537428023)`

Explanation:

`521` is prime and is a factor of `10^52 - 1`.

`10^52 - 1 = 521 * 19193857965451055662188099808061420345489443378119`

So the decimal expansion of `1/521` repeats every `52` digits.

Hence `17 / 5210` will also repeat every `52` digits, after an initial extra `0` after the decimal point due to the factor of `10` in `5210`.

If you long divide `17 / 5210` you will notice that the remainder becomes `17` every `52` steps.

If you divide any positive integer by another, the result will be a decimal expansion that eventually repeats (including the case of repeating `0` - i.e. terminating), since the running remainder must eventually repeat a previous value.