The solution
The sum of all the units we call s.
The sum of the tens consists of the same terms as the sum of the units,
only in a different order. That sum is therefore s.
Similarly, each column adds to s.
The sum of the 120 numbers equals
10000 × s + 1000 × s + 100 × s + 10 × s + 1 × s = 11 111 × s = 271 × 41 × s.
So always the sum is divisible by 41 and by 271.
That's it.
Now probably you yourself can find the next problem.
If you only use 4 digits, then always the sum is divisible by 11 and by 101.
With three digits the dividers are 37 and 3 at least.