24. Returning from the collective farm garden, the boys were enthusiastically discussing the results of their work harvesting carrots. "What are the dimensions of the plot we worked on?" they asked their team leader Vasya. "I lost the measurement records somewhere," he replied, "but I remember that the plot is square and its area in square meters is expressed as a four-digit number, in which all digits are even and different." Based on this information, the boys found the length of the side of the plot and its area. Find them as well.

![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-031.jpg?height=452&width=439&top_left_y=1393&top_left_x=343)

## Search for solutions
24. The problem can be solved simply using a table of squares of numbers. But what if you don't have it at hand? Then we proceed as follows. The smallest four-digit number that is a perfect square is \(1024 (32^2 = 1024)\), and the largest is \(9801 (99^2 = 9801)\). Therefore, if the side length of the plot is \(x\) meters, then \(32 < x < 99\). From these two-digit numbers, we select those whose squares end in an even digit (other than 0). Since squares can end in the digits \(1, 4, 9, 6\), or 5, the sought four-digit number must end in either the digit 4 or 6. We need to look for two-digit numbers (the side length of the square) among the numbers \(32, 34, \ldots, 96, 98\). There are 28 such numbers in total. We exclude those whose squares have an odd digit in the thousands place. Since \(44^2 = 1936\), we exclude 6 numbers from 32 to 44. Since \(55^2 = 3025\) and \(63^2 = 3969\), we exclude three numbers: \(56, 58, 62\). Since \(70^2 = 4900\) and \(77^2 = 5929\), we exclude the numbers \(72, 74, 76\). Since \(84^2 = 7056\) and \(88^2 = 7744\), we exclude \(84, 86, 88\). Since \(94^2 = 8836\) and \(95^2 = 9025\), we exclude \(94, 96, 98\). The remaining numbers for testing are: \(46, 48, 52, 54, 64, 66, 68, 78, 82\), and 92. Among these numbers, only 78 satisfies the problem's requirement: \(78^2 = 6084\). Therefore, the side length of the square is \(78 \text{m}\), and its area is \(6084 \text{m}^2\).

![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-137.jpg?height=400&width=426&top_left_y=445&top_left_x=358)

Divisibility of natural numbers