We begin by noting that    $$\frac{31}{11111} = \frac{31 \times 9}{11111 \times 9} = \frac{279}{99999}.$$We will show that   $$\frac{279}{99999} = 0.\overline{00279},$$so our final answer is $\boxed{5}.$

Proof that $279/99999 = 0.\overline{00279}$:

Let $s = 0.\overline{00279}$. Then multiplying both sides $10^5$ gives   $$10^5 s = 279.\overline{00279}.$$Subtracting the left side by $s$ and the right side by $0.\overline{00279}$ gives   $$99999s = 279,$$so $s = 279/99999$. It follows that $0.\overline{00279} = 279 / 99999,$ as desired.
Question:
When converting $\frac{31}{11111}$ to a decimal, the decimal turns out to be a repeating decimal. How many digits repeat in this repeating decimal?

For example, if you get the repeating decimal $0.\overline{123},$ then your answer should be $3,$ and if you get $0.436\overline{7},$ your answer should be $1.$
Answer:
Let's think step by step.

user
5
When converting $\frac{31}{11111}$ to a decimal, the decimal turns out to be a repeating decimal. How many digits repeat in this repeating decimal?

For example, if you get the repeating decimal $0.\overline{123},$ then your answer should be $3,$ and if you get $0.436\overline{7},$ your answer should be $1.$
test