CEMC Fermat

Number theory Difficulty 2.7 Junior Find the answer

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

Your answer

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

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.

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Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.