In the diagram, a positive integer is to be placed in each of the nine boxes so that the products of the numbers in each row, column, and diagonal are equal. Some of the entries are already filled in. The number of possible values for $N$ is
(A) 4
(D) 6
(B) 15
(C) 9

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-085.jpg?height=249&width=244&top_left_y=2022&top_left_x=1428)
Since all three numbers on the main diagonal are filled in, we can immediately determine what the product of the entries in any row, column or diagonal is, namely $6 \times 12 \times 24=1728$.

We can immediately start to fill in the square by filling in the top-centre, left-centre and bottom-right entries, since we have two entries in each of these rows, columns or diagonals, so the remaining entry is the overall product divided by the two entries already present.

Thus, we obtain

| $N$ | $\frac{1728}{24 N}$ | 24 |
| :---: | :---: | :---: |
| $\frac{1728}{6 N}$ | 12 |  |
| 6 |  | $\frac{1728}{12 N}$ |

Simplifying, we get

| $N$ | $\frac{72}{N}$ | 24 |
| :---: | :---: | :---: |
| $\frac{288}{N}$ | 12 |  |
| 6 |  | $\frac{144}{N}$ |

In a similar way, we can fill in the two remaining entries to get

| $N$ | $\frac{72}{N}$ | 24 |
| :---: | :---: | :---: |
| $\frac{288}{N}$ | 12 | $\frac{1}{2} N$ |
| 6 | $2 N$ | $\frac{144}{N}$ |

Now we are told that each of the nine entries is a positive integer, so each of $N, 2 N, \frac{1}{2} N, \frac{72}{N}, \frac{144}{N}$, and $\frac{288}{N}$ is a positive integer.

Do we need to check each of these conditions?

Well, if $N$ is an integer, then $2 N$ is an integer, so we don't need to check this second condition.

The fact that $\frac{1}{2} N$ is an integer tells us that $N$ has to be an even integer.

The fact that $\frac{72}{N}$ is an integer tells us that $N$ is a factor of 72 .

The fact that $\frac{144}{N}$ is an integer tells us that $N$ is a factor of 144 , but since $N$ is already a factor of 72 and $144=2 \times 72$, then $N$ being a factor of 72 tells us that $N$ is a factor of 144 .

Similarly, $N$ being a factor of 72 tells us that $N$ is a factor of 288 , so $\frac{288}{N}$ is an integer.

In summary, we are looking for positive integers $N$ which are even and factors of 72 .

Writing out the positive factors of 72 , we get $1,2,3,4,6,8,9,12,18,24,36,72$, of which nine are even.