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Problem 319

Number theory Difficulty 2.4 Find the answer CEMC Fermat · Canada · 2021

How many of the 20 perfect squares $1^2, 2^2, 3^2, \dots, 19^2, 20^2$ are divisible by 9?

$2$
$3$
$4$
$5$
$6$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

A perfect square is divisible by 9 exactly when its square root is divisible by 3.

In other words, $n^2$ is divisible by 9 exactly when $n$ is divisible by 3.

In the list $1, 2, 3, \ldots, 19, 20$ , there are 6 multiples of 3.

Therefore, in the list $1^2, 2^2, 3^2, \ldots, 19^2, 20^2$ , there are 6 multiples of 9.

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