4. Inside the rectangle $A B C D$, there are 9 equal rectangles as shown in the figure. All the gaps between the rectangles are equal - the gap between two smaller rectangles and the gap between a smaller rectangle and the side of rectangle $A B C D$. Calculate the area of the shaded part if the area of rectangle $A B C D$ is $697 \mathrm{~cm}^{2}$. The lengths of the sides of all rectangles and the length of the gaps are natural numbers in centimeters.

![](https://cdn.mathpix.com/cropped/2024_06_03_84104ab88f15e63b3af1g-1.jpg?height=339&width=737&top_left_y=2052&top_left_x=728)
First method:

Let's consider the image and mark the appropriate gaps according to the conditions of the problem:

![](https://cdn.mathpix.com/cropped/2024_05_30_c165b3842bf96502feaeg-04.jpg?height=497&width=1123&top_left_y=420&top_left_x=475)

Let \(a\) and \(b\) be the lengths of the adjacent sides of the smaller rectangle, \(a > b\), and \(x\) be the width of the gap. Then the lengths of the sides of rectangle \(ABCD\) are \(|AB| = 3 \cdot a + 4 \cdot x\) and \(|BC| = 3 \cdot b + 4 \cdot x\), and the area of rectangle \(ABCD\) is \(|AB| \cdot |BC|\), so the equation \((3 \cdot a + 4 \cdot x) \cdot (3 \cdot b + 4 \cdot x) = 697\) holds. If we factorize the number 697 into prime factors, we get \(697 = 41 \cdot 17\). From the condition \(a > b\), it follows that \(3 \cdot a > 3 \cdot b\) or \(3 \cdot a + 4 \cdot x > 3 \cdot b + 4 \cdot x\). Therefore, \(|AB| = 3 \cdot a + 4 \cdot x = 41\) and \(|BC| = 3 \cdot b + 4 \cdot x = 17\). In the expression \(3 \cdot b + 4 \cdot x = 17\), the term \(4 \cdot x\) is always even, and the sum is odd, so the number \(b\) must be an odd number less than 7 (\(3 \cdot 7 = 21, 21 > 17\)).

For \(b = 1\), we have \(3 + 4 \cdot x = 17\) or \(4 \cdot x = 14\). However, the solution to this equation is not a natural number. For \(b = 3\), we have \(9 + 4 \cdot x = 17\) or \(4 \cdot x = 8\), so \(x = 2\).

For \(b = 5\), we have \(15 + 4 \cdot x = 17\) or \(4 \cdot x = 2\). However, the solution to this equation is not a natural number.

Substituting \(x = 2\) into the equation \(3 \cdot a + 4 \cdot x = 41\), we get \(3 \cdot a + 8 = 41\), \(3 \cdot a = 33\), so \(a = 11\).

The lengths of the sides of the smaller rectangle are \(11 \, \text{cm}\) and \(3 \, \text{cm}\), so the area of one smaller rectangle is \(33 \, \text{cm}^2\). The area of all nine smaller rectangles is \(297 \, \text{cm}^2\).

The area of the shaded part is the difference between the area of rectangle \(ABCD\) and the area of the nine smaller rectangles: \(697 - 297 = 400 \, \text{cm}^2\).