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Problem 433

Number theory Difficulty 2.7 Multiple choice CEMC Gauss (Grade 7) · Canada · 2025

Ten students each receive a card numbered with a different
integer from $10$ to $19$ . The students are each given the
checklist shown and they check off each box that describes their
number.

☐ Odd Number
☐ Even Number
☐ Prime
Number
☐ Composite
Number
☐ Perfect
Square

How many students check off exactly two boxes?

Pick one

Official solution

Each integer is either an odd number or it is an even
number.

Thus, each of the $10$ students will
check off exactly one box from the first pair of boxes.

Similarly, every integer greater than $1$ is either a prime number or it is a
composite number, and so each of the $10$ students will check off exactly one
box from the second pair of boxes.

Thus, every student checks off exactly $2$ of the first $4$ boxes.

This means that every student checks off exactly $2$ of the $5$ boxes if their card is not
numbered with a perfect square, and they check off exactly $3$ of the $5$ boxes if their card is
numbered with a perfect square.

The card numbered $16$ is the only
card numbered with a perfect square, and so each of the remaining $10-1=9$ students have a card that is not
numbered with a perfect square.

Thus, there are $9$ students that
check off exactly two boxes.

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