list=[];

for n=0:8191

  ns=dec2bin(n,13);

  N=str2double(char(ns+7));

  m=mod(N,3^13);

  if ~ismember(mod(N,3^13),list)

    list(end+1)=mod(N,3^13);

  end

  % disp(mod(N,3^13))

  if mod(N,3^13)==0

    disp(N)

  end

end

length(list)

min(list)

max(list)

3^13-max(list)

checks all such numbers and finds no such divisible number.

The output 

ans =

        8100

ans =

   303

ans =

     1588566

ans =

        5757

        

shows there are 8100 different values mod 3^13, the smallest being 303 and the largest 1588566, which is 5757 short of being congruent to zero.

The numbers that lead to the lowest and highest mod values are  8877888777777 and 8788888878888.