2024 AMC 10B 第 6 题

Question

一个长方形的边长都是整数，面积为

2024

。它的周长最小可能是多少？

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

对固定面积，周长在两条边尽量接近时最小。

For a fixed area, the perimeter is smallest when the two sides are as close to equal as possible

2024 = 2^3 \cdot 11 \cdot 23

；寻找接近

\sqrt{2024} \approx 45

的因数对。

2024 = 2^3 \cdot 11 \cdot 23;

look for a factor pair near

\sqrt{2024} \approx 45

若长和宽为

\ell

和

w

，周长为

2(\ell + w)

，在

\ell w = 2024

且两边尽量接近时最小。分解

2024 = 2^3 \cdot 11 \cdot 23

。最接近

\sqrt{2024} \approx 45

的因数对是

44 \times 46

，周长为

2(44 + 46) = 180

。因此正确答案是 B。

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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