1. The perimeter of a rectangle is 40 cm, and its area does not exceed $40 \mathrm{~cm}^{2}$. The length and width of the rectangle are expressed as natural numbers. By how much will the area of the rectangle increase if its perimeter is increased by 4 cm?
1. Let's determine what the semi-perimeter of the rectangle is equal to $(40: 2=20)$. Since the sum of the length and width is 20, the sides can be 19 and 1, 18 and 2, 17 and 3, etc. Only the first two cases satisfy the second condition of the problem: 19 and $1(19 \times 1=19)$, 18 and 2 $(18 \times 2=36)$.

If the perimeter is increased by 4, then the semi-perimeter of the new rectangle will be 22 (21 and 1, 20 and 2, 19 and 3, 18 and 4). The area of the rectangle can increase by:

$21 \times 1-19 \times 1=2 \mathrm{~cm}^{2} ; 20 \times 2-19 \times 1=21 \mathrm{~cm}^{2} ; 19 \times 3-19 \times 1=38 \mathrm{~cm}^{2}$; $20 \times 2-18 \times 2=4 \mathrm{~cm}^{2} ; 19 \times 3-18 \times 2=21 \mathrm{~cm}^{2} ; 18 \times 4-18 \times 2=36 \mathrm{~cm}^{2}$.

Answer: the area will increase by $2, 4, 21, 36$ or $38 \mathrm{~cm}^{2}$.