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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.