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 = 10^{P} X – X = (10^{P}-1) X = (10^{P}-1) (m/n).

Thus R*n = (10^{P}-1)*m is an integer. Since (10^{P}-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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