Repeating Digits

You already know that the decimal expansion of a rational number eventually repeats or terminates (which can be viewed as a repeating 0).

But I tell you something that perhaps you did not know: if the denominator of that rational number is not divisible by 3, then the repeating part of its decimal expansion is an integer divisible by nine!

Example:

  • 1/7 = .142857142857… has repeating part 142857. This is divisible by 9.
  • 41/55 = .7454545… has repeating part 45. This is divisible by 9.

The Math Behind the Fact:
This rather curious fact can be shown easily. If the rational X is purely repeating of period P and repeating part R, then

R = 10P X – X = (10P-1) X = (10P-1) (m/n).

Thus R*n = (10P-1)*m is an integer. Since (10P-1) is divisible by 9, if n is not divisible by 3, then R must be. If you like these fun deductions, you may enjoy a course in number theory!

How to Cite this Page: 
Su, Francis E., et al. “Repeating Digits.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

Fun Fact suggested by:
Arthur Benjamin

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