Your birthyear has 4 digits let it be abcd
I mean
1000a+100b+10c+d that is the year you were born.
now if I add the 4 digits i get a+b+c+d
so doing the subtraction I get
1000a+100b+10c+d -(a+b+c+d)
=1000a+100b+10c+d -a-b-c-d
=999a+99b+9c
=9(111a+11b+c)
Which is obviously divisable by 9 since 9 is one of the factors 
Isn't mathematics fun 