a time to repeat
Consider the following decimal expansions:
$$\begin{aligned} \textstyle \frac{1}{ 1 } &= 1.0 & \textstyle \frac{1}{ 11 } &= 0.[09] & % 0.09090909090909091 \\ \textstyle \frac{1}{ 21 } &= 0.[04761\,9] \\ % 0.047619047619047616 \\ % \textstyle \frac{1}{ 2 } &= 0.5 & \textstyle \frac{1}{ 12 } &= 0.08[3] & % 0.08333333333333333 \\ \textstyle \frac{1}{ 22 } &= 0.0[45] \\ % 0.045454545454545456 \\ % \textstyle \frac{1}{ 3 } &= 0.[3] & % 0.3333333333333333 \\ \textstyle \frac{1}{ 13 } &= 0.[07692\,3] & % 0.07692307692307693 \\ \textstyle \frac{1}{ 23 } &= 0.[04347\,82608\,69565\,21739\,13] \\ % 0.043478260869565216 \\ % \textstyle \frac{1}{ 4 } &= 0.25 & \textstyle \frac{1}{ 14 } &= 0.0[71428\,5] & % 0.07142857142857142 \\ \textstyle \frac{1}{ 24 } &= 0.041[6] \\ % 0.041666666666666664 \\ % \textstyle \frac{1}{ 5 } &= 0.2 & \textstyle \frac{1}{ 15 } &= 0.0[6] & % 0.06666666666666667 \\ \textstyle \frac{1}{ 25 } &= 0.04 \\ % \textstyle \frac{1}{ 6 } &= 0.1[6] & % 0.16666666666666666 \\ \textstyle \frac{1}{ 16 } &= 0.0625 & \textstyle \frac{1}{ 26 } &= 0.0[38461\,5] \\ % 0.038461538461538464 \\ % \textstyle \frac{1}{ 7 } &= 0.[14285\,7] & % 0.14285714285714285 \\ \textstyle \frac{1}{ 17 } &= 0.[05882\,35294\,11764\,7] & % 0.058823529411764705 \\ \textstyle \frac{1}{ 27 } &= 0.[037] \\ % 0.037037037037037035 \\ % \textstyle \frac{1}{ 8 } &= 0.125 & \textstyle \frac{1}{ 18 } &= 0.0[5] & % 0.05555555555555555 \\ \textstyle \frac{1}{ 28 } &= 0.035[71428\,5] \\ % 0.03571428571428571 \\ % \textstyle \frac{1}{ 9 } &= 0.[1] & % 0.1111111111111111 \\ \textstyle \frac{1}{ 19 } &= 0.[05263\,15789\,47368\,421] & % 0.05263157894736842 \\ \textstyle \frac{1}{ 29 } &= 0.[03448\,27586\,20689\,65517\,24137\,931] \\ % 0.034482758620689655 \\ % \textstyle \frac{1}{ 10 } &= 0.1 & \textstyle \frac{1}{ 20 } &= 0.05 & \textstyle \frac{1}{ 30 } &= 0.0[3] \\ % 0.03333333333333333 \\ \end{aligned}$$
Each of these either terminates, like $\frac18 = 0.125$, or repeats, like $\frac{1}{27} = 0.[037]$. What determines the length of these repeating sequences?
