11. $[\mathbf{8}] \mathrm{Al}$ has a rectangle of integer side lengths $a$ and $b$, and area 1000 . What is the smallest perimeter it could have?
Answer: 130 To minimize the sum of the side lengths, we need to keep the height and width as close as possible, because the square has the smallest perimeter of all rectangles with a fixed area. So, 40 and 25 multiply to 1000 and are as close as possible - the $40 \times 25$ rectangle has perimeter 130 .