# What is the decimal representation of 17/5210 ?

Oct 13, 2015

$17 \div 5210 = .003262955854$

#### Explanation:

Using a calculator, $17 \div 5210 = .003262955854$.

Oct 13, 2015

$\frac{17}{5210} = 0.0 \overline{0326295585412667946257197696737044145873320537428023}$

#### Explanation:

$521$ is prime and is a factor of ${10}^{52} - 1$.

${10}^{52} - 1 = 521 \cdot 19193857965451055662188099808061420345489443378119$

So the decimal expansion of $\frac{1}{521}$ repeats every $52$ digits.

Hence $\frac{17}{5210}$ 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 $\frac{17}{5210}$ 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.