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Problem 170

Number theory Difficulty 1.4 Multiple choice CEMC Gauss (Grade 7) · Canada · 2025

The numbers $2$ , $0$ , $2$ , $5$
are repeated to form the pattern $2$ ,
$0$ , $2$ , $5$ , $2$ , $0$ , $2$ , $5$ , $\ldots$

If a total of $50$ numbers are
written, how many times will the number $5$ appear?

Pick one

Official solution

The pattern repeats in blocks of the $4$ numbers $2$ , $0$ , $2$ , $5$ . Since $50=12\times4+2$ , then the $50$ numbers in the pattern contain $12$ complete blocks of $2$ , $0$ , $2$ , $5$
followed by the first $2$ numbers in
the block, $2$ , $0$ .

The number $5$ appears once in each
of the $12$ blocks and does not
appear as the 49th or 50th number. Thus, the number $5$ appears $12$ times.

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