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 $5?$

$8$

$9$

$10$

$11$

$12$

Answer: A

Concepts:perfect square divisibility

Difficulty rating: 1050

Solution:

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.

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