2024 AMC 10B Problem 6

Attempt Problem 6 of the 2024 AMC 10B 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 2024 AMC 10B solutions, or check the answer key.

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

A rectangle has integer side lengths and an area of $2024.$ What is the least possible perimeter of the rectangle?

$160$

$180$

$222$

$228$

$390$

Answer: B

Concepts:factor perimeter optimization

Difficulty rating: 1200

Solution:

The perimeter $2(\ell + w)$ with $\ell w = 2024$ is smallest when $\ell$ and $w$ are as close together as possible. Factor $2024 = 2^3 \cdot 11 \cdot 23.$ The divisor pair nearest $\sqrt{2024} \approx 45$ is $44 \times 46,$ which gives perimeter $2(44 + 46) = 180.$ Therefore, the answer is B.

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