reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th11:
  n divides 27 implies n = 1 or n = 3 or n = 9 or n = 27
  proof
    assume
A1: n divides 27;
    then n <= 27 by INT_2:27;
    then
A2: n = 0 or ... or n = 27;
    now
      (2*13+1) mod 2 = 1 mod 2 by NAT_D:21
      .= 1 by NAT_D:24;
      hence not 2 divides 27;
      (4*6+3) mod 4 = 3 mod 4 by NAT_D:21
      .= 3 by NAT_D:24;
      hence not 4 divides 27 by INT_1:62;
      (5*5+2) mod 5 = 2 mod 5 by NAT_D:21
      .= 2 by NAT_D:24;
      hence not 5 divides 27 by INT_1:62;
      (6*4+3) mod 6 = 3 mod 6 by NAT_D:21
      .= 3 by NAT_D:24;
      hence not 6 divides 27 by INT_1:62;
      (7*3+6) mod 7 = 6 mod 7 by NAT_D:21
      .= 6 by NAT_D:24;
      hence not 7 divides 27 by INT_1:62;
      (8*3+3) mod 8 = 3 mod 8 by NAT_D:21
      .= 3 by NAT_D:24;
      hence not 8 divides 27 by INT_1:62;
      (10*2+7) mod 10 = 7 mod 10 by NAT_D:21
      .= 7 by NAT_D:24;
      hence not 10 divides 27 by INT_1:62;
      (11*2+5) mod 11 = 5 mod 11 by NAT_D:21
      .= 5 by NAT_D:24;
      hence not 11 divides 27 by INT_1:62;
      (12*2+3) mod 12 = 3 mod 12 by NAT_D:21
      .= 3 by NAT_D:24;
      hence not 12 divides 27 by INT_1:62;
      (13*2+1) mod 13 = 1 mod 13 by NAT_D:21
      .= 1 by NAT_D:24;
      hence not 13 divides 27 by INT_1:62;
      (14*1+13) mod 14 = 13 mod 14 by NAT_D:21
      .= 13 by NAT_D:24;
      hence not 14 divides 27 by INT_1:62;
      (15*1+12) mod 15 = 12 mod 15 by NAT_D:21
      .= 12 by NAT_D:24;
      hence not 15 divides 27 by INT_1:62;
      (16*1+11) mod 16 = 11 mod 16 by NAT_D:21
      .= 11 by NAT_D:24;
      hence not 16 divides 27 by INT_1:62;
      (17*1+10) mod 17 = 10 mod 17 by NAT_D:21
      .= 10 by NAT_D:24;
      hence not 17 divides 27 by INT_1:62;
      (18*1+9) mod 18 = 9 mod 18 by NAT_D:21
      .= 9 by NAT_D:24;
      hence not 18 divides 27 by INT_1:62;
      (19*1+8) mod 19 = 8 mod 19 by NAT_D:21
      .= 8 by NAT_D:24;
      hence not 19 divides 27 by INT_1:62;
      (20*1+7) mod 20 = 7 mod 20 by NAT_D:21
      .= 7 by NAT_D:24;
      hence not 20 divides 27 by INT_1:62;
      (21*1+6) mod 21 = 6 mod 21 by NAT_D:21
      .= 6 by NAT_D:24;
      hence not 21 divides 27 by INT_1:62;
      (22*1+5) mod 22 = 5 mod 22 by NAT_D:21
      .= 5 by NAT_D:24;
      hence not 22 divides 27 by INT_1:62;
      (23*1+4) mod 23 = 4 mod 23 by NAT_D:21
      .= 4 by NAT_D:24;
      hence not 23 divides 27 by INT_1:62;
      (24*1+3) mod 24 = 3 mod 24 by NAT_D:21
      .= 3 by NAT_D:24;
      hence not 24 divides 27 by INT_1:62;
      (25*1+2) mod 25 = 2 mod 25 by NAT_D:21
      .= 2 by NAT_D:24;
      hence not 25 divides 27 by INT_1:62;
      (26*1+1) mod 26 = 1 mod 26 by NAT_D:21
      .= 1 by NAT_D:24;
      hence not 26 divides 27 by INT_1:62;
    end;
    hence thesis by A1,A2;
  end;
