3. In a $100 \times 25$ rectangular table, each cell is filled with a non-negative real number, with the number in the $i$-th row and $j$-th column being $x_{i, j}(i=1,2, \cdots, 100 ; j=1,2, \cdots, 25)$ (Table 1). Then, the numbers in each column of Table 1 are rearranged in descending order from top to bottom as $x_{1, j}^{\prime} \geqslant x_{2, j}^{\prime} \geqslant \cdots \geqslant x_{100, j}^{\prime}(j=1,2, \cdots, 25)$ (Table 2). Find the smallest natural number $k$ such that if the numbers in Table 1 satisfy $\sum_{j=1}^{25} x_{i, j} \leqslant 1(i=1,2, \cdots, 100)$, then when $i \geqslant k$, in Table 2, $\sum_{j=1}^{25} x_{i, j}^{\prime} \leqslant 1(i=1,2, \cdots, 100)$ holds.
Let $x_{1,1}=x_{2,1}=x_{3,1}=x_{4,1}=x_{5,2}=x_{6,2}=x_{7,2}=x_{8,2}=\cdots=x_{97,25}=x_{98,25}=$ $x_{99,25}=x_{100,25}=0$, the rest of the elements are $\frac{1}{24}$. Then the sum of 25 numbers in each row is $0+24 \times \frac{1}{24}=1$. At this point, each column of Table 1 has exactly 4 zeros, so after adjustment, the last 4 rows of Table 2 are all 0, and the numbers in the first 96 rows are all $\frac{1}{24}$, then the sum of 25 numbers in each row is $25 \times \frac{1}{24}=\frac{25}{24}>1 \Rightarrow k \geqslant 97$.
The following proves that the minimum value of $k$ is 97. Since the last 3 rows (the 98th, 99th, and 100th) can only accommodate 75 elements, there must be a row (let's say the $r$-th row) in Table 1, all of its 25 elements are located in the first 97 rows after adjustment in Table 2. Otherwise, if each row has at least one element falling into the last 3 rows, then at least 100 elements would fall into the last 3 rows, which is a contradiction. Thus, $x_{97, j}^{\prime} \leqslant x_{r, j}\left(x_{r, j}\right.$ is still in the $j$-th column of Table 2, the row number is not necessarily $r$, but the row number $\leqslant 97)$. Therefore, for any $i \geqslant 97$, we have $x_{i, j}^{\prime} \leqslant x_{97, j}^{\prime} \leqslant x_{r, j}$ $\Rightarrow \sum_{j=1}^{25} x_{i, j}^{\prime} \leqslant \sum_{j=1}^{25} x_{r, j} \leqslant 1$. So the minimum value of $k$ is 97.