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Question

When converting 3111111 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 0.123, then your answer should be 3, and if you get 0.4367, your answer should be 1.

Accepted Answer

We begin by noting that 3111111 = 31 911111 9 = 27999999.We will show that 27999999 = 0.00279,so our final answer is 5. Proof that 279/99999 = 0.00279: Let s = 0.00279. Then multiplying both sides 10^5 gives 10^5 s = 279.00279.Subtracting the left side by s and the right side by 0.00279 gives 99999s = 279,so s = 279/99999. It follows that 0.00279 = 279 / 99999, as desired.