Given 361, find the smallest positive integer $n$, such that 16 numbers can be selected from $1,2, \cdots, n$ and filled into a $4 \times 4$ grid so that the product of the numbers in each row and each column is equal.
The smallest value of the positive integer $n$ is 27.
First, when $n=27$, a valid arrangement is shown in Figure 3.
Next, we prove that the smallest value of $n$ is 27.
Consider the prime factorization of the product $P$ of each row or column.
(1) If $P$ has only one prime factor, then $n \geqslant 2^{15} > 27$.
(2) If $P$ has exactly two prime factors, to minimize the largest number, these two prime factors should be as small as possible, taken as 2 and 3. In this case, at most 16 numbers can be selected from $1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, \cdots$, clearly, $n > 27$.
(3) If $P$ has exactly three prime factors, to minimize the largest number, these three prime factors should be as small as possible, taken as 2, 3, and 5. In this case, at most 16 numbers can be selected from $1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, \cdots$, and only 25 has two 5s, which cannot be selected, clearly, $n \geqslant 27$.
(4) If $P$ has more than three prime factors, then $n \geqslant 4 \times 7 = 28$.
In summary, the smallest value of the positive integer $n$ is 27.
(Song Hongjun, Fuyang High School, Zhejiang, 311400)