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Problem 461

Number theory Difficulty 2.7 Multiple choice CEMC Gauss (Grade 8) · Canada · 2022

The positive factors of 6 are 1, 2, 3, and 6. There are two
perfect squares less than 100 that have exactly five positive factors.
What is the sum of these two perfect squares?

Pick one

Official solution

We begin by constructing the three diagonals inside the smaller
squares that were missing from the diagram given in the question, as
shown.

[[IMAGE0]]

The two diagonals inside each of these 4 smaller squares divide each
smaller square into 4 identical triangles having equal area.

Thus, square $ABCD$ is divided into
$4\times4=16$ such triangles.

Since 7 of these triangles are shaded, then the fraction of $ABCD$ that is shaded is $\frac{7}{16}$ .

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