A test for Belgian-ity

[Eugene McDonnell]

More or less empirically, I have found that I can make an easy test to 
see if a given positive integer is Belgian; I'd appreciate it if 
someone more versed in mathematics could prove or disprove the 
following, before I pollute the Encyclopedia with an incorrect comment 
for A106039:

To test the Belgian-ity of a positive integer n, find s, the cumulative 
sum of its digits, and m the sum of its digits; then n is Belgian if 
there is a 0 in any of the differences of n minus the items of s, mod 
m. For example, if n is 176 then s is 1 8 14, and m is 14; 176 - 1 8 14 
is 175 168 162, and the residues mod(14) are 7 0 8, so 176 is Belgian. 
Contrariwise, if n is 177, s is 1 8 15, and m is 15; 177 - 1 8 15 is 
176 169 162, and the residues, mod(15) are 11 4 12, so 177 is not 
Belgian.

A more exotic example:

Let n be 1234567898765; then s is 1 3 6 10 15 21 28 36 45 53 60 66 71, 
and m is 71.

    (1234567898765 - 1 3 6 10 15 21 28 36 45 53 60 66 71) mod(71) is:

20 18 15 11 6 0 64 56 47 39 32 26 21

which contains a 0, so n is Belgian.

Eugene McDonnell