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Problem 78

Number theory Difficulty 1.2 Multiple choice CEMC Gauss (Grade 7) · Canada · 2023

Sarah multiplied an integer by itself. Which of the following
could be the result?

Pick one

Official solution

Since $5\times5=25$ and each
of the given answers is greater than 25, then the integer that Sarah
multiplied by itself must have been greater than 5.

Further, $6\times6=36$ and each of
the given answers is less than or equal to 36.

Thus, of the given answers, only 36 could be the result of multiplying
an integer by itself.

Alternatively, we may have noted that the result of multiplying an
integer by itself is a perfect square, and of the answers given, 36 is
the only perfect square.

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