Question 75
Define a multiplication table to be a rectangular array in which every row is labeled with a different positive integer, every column is labeled with a different positive integer, and every cell is labeled with the product of its row and column numbers, for instance:
| $\times$ | 2 | 6 | 4 | 3 |
| :---: | :---: | :---: | :---: | :---: |
| 2 | 4 | 12 | 8 | 6 |
| 1 | 2 | 6 | 4 | 3 |
| 3 | 6 | 18 | 12 | 9 |
The above table is $3 \times 4$ but contains only 8 distinct products. What is the minimum number of distinct products in a $2012 \times 2012$ multiplication table?
| $\times$ | 2 | 6 | 4 | 3 |
| :---: | :---: | :---: | :---: | :---: |
| 2 | 4 | 12 | 8 | 6 |
| 1 | 2 | 6 | 4 | 3 |
| 3 | 6 | 18 | 12 | 9 |
The above table is $3 \times 4$ but contains only 8 distinct products. What is the minimum number of distinct products in a $2012 \times 2012$ multiplication table?