Problem of the Week Problem C and Solution Again and Again

Problem

The fraction $$\frac{1}{7}$$ is equal to the repeating decimal $$0.\overline{142857}$$.

Which digit occurs in the 2023rd place after the decimal point?

Solution

The digits after the decimal point occur in repeating blocks of the $$6$$ digits $$142857$$.

Since $$\frac{2023}{6}=337.1\overline{6}=337 \frac{1}{6}$$, it follows that the 2023rd digit after the decimal point occurs after $$337$$ complete repeating blocks of the $$6$$ digits.

In $$337$$ complete repeating blocks, there are $$337\times 6=2022$$ digits in total. The 2023rd digit is then the next digit. This corresponds to the first digit in the repeating block, which is $$1$$.

Therefore, the digit $$1$$ occurs in the 2023rd place after the decimal point.