number 19

Code: PU 
P=100 (k=81=5 (mod 19) )
Formulae:
Number N is divisible by 19 iff 5*b+a is divisible by 19

Example:
Number N = 98 76 54 32 16 = 3 0 16 13 16 =
5(3 0 16 13)+16 =
5(5( 3 0 16+13)+16 =
5(5(5(  3 0+16)+13)+16 =
5(5(5(5*  3+0)+16)+13)+16 =
5(5(5*    15+16)+13)+16 =
5(5*      15+13)+16
5*        12+16 = 0 (mod 19)
is divisible by 19.



Code: PD
P=10, (k=9)
Formulae:
Number N is divisible by 19 iff -9b+a = 10b+a is divisible by 19

Example:
Number N = 54391
-9(      5439)+1 =
-9(-9(    543+9)+1 =
-9(-9(-9(  54+3)+9)+1 =
-9(-9(-9(-9*5+4)+3)+9)+1 =
-9(-9(-9*(-3)+3)+9)+1 =
-9(-9*     11+9)+1
-9*         5+1 = 13 (mod 19)
isn't divisible by 19.



Code: SF  2*10-1*19=1
2(10b+a) = b+2a (mod 19) 
Formulae:
Number N is divisible by 19 iff b+2a is divisible by 19

Example:
Number N = 68932
6893+2*2 = 6897
 689+2*7 =703
  70+2*3 =76
   7+2*6 = 19
is divisible by 19



Code: SF-  17*10-9*19=-1
17(10b+a) = -b+17a = -(b+2a) (mod 19)
It is the same as above