#
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 2023^{rd} 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 2023^{rd} 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 2023^{rd} 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 2023^{rd} place after the decimal point.