Question about determinants (restated)

Look at this:
1.
The two numbers 21 and 12 are divisible by 3. The determinant

| 2 1 |
| 1 2 |

is equal to 2*2-1*1=3, which is divisible by 3.

Or for another example, the three numbers 112, 763, and 959 are divislbe by 7.

The determinant

|1 1 2|
|7 6 3|
|9 5 9|

= 6*9 + 3*9 + 2*7*5 - 9*6*2 - 5*3 - 7*9 = 54 + 27 + 70 - 108 - 15 - 63 = -35 which is also divisible by 7.

The question is this: does this always work?