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, 2015

Number theory Difficulty 2.6 Junior Find the answer Canada

When expressed as a repeating decimal, the fraction $\frac{1}{7}$ is written as $0.142857142857\dots$ (The 6 digits 142857 continue to repeat.) The digit in the third position to the right of the decimal point is 2. In which one of the following positions to the right of the decimal point will there also be a 2?

Pick one

Solutions — 2

Solution 1

Since the number of digits that repeat is 6, then the digits $142857$ begin to repeat again after 120 digits (since $120=6\times20$ ).
That is, the $121^{st}$ digit is a 1, the $122^{nd}$ digit is a 4, and the $123^{rd}$ digit is a 2.

Solution 2

The original cube (before the corner was cut off) had 12 edges.

Cutting off the corner does not eliminate any of the 12 edges of the original cube.
However, cutting off the corner does add 3 edges that were not present originally, the 3 edges of the new triangular face. Since no edges of the original cube were lost, but 3 new edges were created, then the new solid has $12+3=15$ edges.

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