2023 AMC 12A Problem 3

Below is the professionally curated solution for Problem 3 of the 2023 AMC 12A, 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.

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Concepts:perfect squaredivisibility

Difficulty rating: 1130

3.

How many positive perfect squares less than 20232023 are divisible by 5?5?

88

99

1010

1111

1212

Solution:

A perfect square is divisible by 55 only if its root is, so the squares are (5k)2=25k2.(5k)^2=25k^2.

Since 442=1936<2023<2025=452,44^2=1936\lt 2023\lt 2025=45^2, the root can be 5,10,,40,5,10,\ldots,40, which is k=1k=1 through 8.8.

Thus, the correct answer is A.

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