Problem:
A rectangle has integer length sides and an area of 2024 . What is the least possible perimeter of the rectangle?
Answer Choices:
A. 160
B. 180
C. 222
D. 228
E. 390
Solution:
Note that 2024=44⋅46. A 44×46 rectangle will have perimeter 2(44+46)=180. It is straightforward to check the other possible dimensions to show that this gives the rectangle with the least possible perimeter:
- 23×88 gives a perimeter of 2(23+88)=222.
- 22×92 gives a perimeter of 2(22+92)=228.
- 11×184 gives a perimeter of 2(11+184)=390.
- If one of the dimensions is 1,2,4, or 8 , then the other dimension is greater than 200 , yielding rectangles with greater perimeters.
Thus the least possible perimeter is (B)180​ .
The problems on this page are the property of the MAA's American Mathematics Competitions