Determine the total number of concatenated 3-digit positive integers of the form ABC, such that each of their nonzero digits A, B, and C satisfies this relationship:

           .     .     .         
  ___    0.A + 0.B + 0.C
0.ABC = ------------------
                 3

Notes:

.     ---
    0.ABC = 0.ABCABCABC.......
  .
0.A = 0.AAAA.......

Computer program/excel solver solutions are welcome- but a semi-analytic (p&p with a hand-calculator) methodology is preferred.