2023 AMC 12A Problem 3

Attempt Problem 3 of the 2023 AMC 12A 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 12A 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 is divisible by

5

exactly when its root is a multiple of

5

The squares are

(5k)^2

; find how many satisfy

(5k)^2\lt 2023

A perfect square is divisible by

5

only if its root is, so the squares are

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

Since

44^2=1936\lt 2023

\lt 2025=45^2,

the root can be

5,10,\ldots,40,

which is

k=1

through

8.

Thus, the correct answer is A.

Problem 3 in other years