2023 AMC 10A Problem 3

Attempt Problem 3 of the 2023 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

How many positive perfect squares less than

2023

are divisible by

5?

8

9

10

11

12

A perfect square divisible by

5

must be divisible by

25

Write the square as

(5k)^2 = 25k^2 \lt 2023

and count the valid

k

If a perfect square is divisible by

5,

it’s divisible by

25,

so it looks like

(5k)^2 = 25k^2.

We need

25k^2 \lt 2023,

i.e.

k^2 \lt 80.9.

That allows

k = 1, 2, \ldots, 8,

which is

8

squares. Thus, A is the correct answer.

Problem 3 in other years