If I tell you that each of the four numbers 1242,1426,2070,and 1564
is divisible by 23(trust me!), could you with no explicit calculation prove that the determinant of the 4x4 matrix
1 2 4 2
1 4 2 6
1 5 6 4
2 0 7 0
is also divisible by 23?

Multiply
the 1st, 2^{nd} and 3^{rd} columns by 1000, 100 and 10
respectively
and add the results to the fourth column. This will not change the
value of the determinant:

1 2 4 1242
1421426
1561564
2072070:

23 is then a common factor of all elements in column 4 and therefore
a factor of the determinant.