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Problem 130

Number theory Difficulty 1.3 Multiple choice CEMC Pascal · Canada · 2023

The decimal representation of $\frac{1}{11}$ is $0.09090909...$ .

Another way to write this decimal representation is $0.\overline{09}$ .

Similarly, $0.\overline{125}$
represents the number $0.125125125\ldots$ .

The decimal representation of $\frac{1}{7}$ is $0.\overline{142857}$ .

In the decimal representation of $\frac{1}{7}$ , the 100th digit to the right
of the decimal is

Pick one

Official solution

The digits to the right of the decimal place in the decimal
representation of $\frac{1}{7}$ occur
in blocks of 6, repeating the block of digits 142857.

Since $16 \times 6 = 96$ , then the
96th digit to the right of the decimal place is the last in one of these
blocks; that is, the 96th digit is 7.

This means that the 97th digit is 1, the 98th digit is 4, the 99th digit
is 2, and the 100th digit is 8.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.