number 31

Code: PU  
P=100 (k=69=7 (mod 31) )

Formulae:
Number N is divisible by 31 iff 7*b+a is divisible by 31

Example:
Number N = 98 76 54 32 08 = 5 14 23 1 8 =
7(  5 14 23 1)+8 =
7(7(5 14 23+1)+8 =
7(7(7( 5 14+23)+1)+8 =
7(7(7(7*  5+14)+23)+1)+8 =
7(7(7*   18+23)+1)+8 =
7(7*     25+1)+8 =
7*       21+8 = 0 (mod 31)
is divisible by 31.



Code: SF  28*10-9*31=1
28(10b+a) = b+28a = b-3a (mod 31)
Formulae:
Number N is divisible by 31 iff b-3a is divisible by 31

Example:
Number N = 68931
6893-3*1 = 6890
 689-3*0 =689
  68-3*9 = 41
   4-3*1 = 1
isn't divisible by 31



Code: SF-  3*10-1*31=-1
3(10b+a) = -b+3a = -(b-3a) (mod 31)
It is the same as above.

number 30

Code: CR
P = 3*10

Formulae: 
Number N is divisible by 30 iff least significant digit is 0

Example:
Convert decimal number N = 8512160 into base P = 3*10=30, first step is enough
[8 5 1 2 1 6 | 0] = [6 24 9 21 9 24 | 20] = 3*[2 8 3 7 3 8] 20 
Number N isn't divisible by 30



Code: PU  
P=100 (k=70=10 (mod 30) )
This case method doesn't work - the formulae wishes to check a number N:
Number N is divisible by 30 iff N = 10*b+a is divisible by 30

number 29

Code: PU  
P=100 (k=71=13 (mod 29) )

Formulae:
Number N is divisible by 29 iff 13*b+a is divisible by 29

Example:
Number N = 98 76 54 32 13 = 11 18 25 3 13 =
13( 11 18 25 3)+13 =
13(13(11 18 25+3)+13 =
13(13(13(11 18+25)+3)+13 =
13(13(13(11*17+18)+25)+3)+13 =
13(13(13*    2+25)+3)+13 =
13(13*      22+3)+13 =
13*         28+13 = 0 (mod 29)
is divisible by 29.



Code: SF  3*10-1*29=1
3(10b+a) = b+3a (mod 29)
Formulae:
Number N is divisible by 29 iff b+3a is divisible by 29

Example:
Number N = 68931
6893+3*1 = 6896
 689+3*6 = 707
  70+3*7 = 91
   9+3*1 = 12
isn't divisible by 29



Code: SF-  26*10-9*29=-1
26(10b+a) = -b+26a = -(b+3a) (mod 29)
It is the same as above.

number 28

Code: PU  P=100 (k=72=16 (mod 28) )
Formulae:
Number N is divisible by 28 iff 16b+a = -12b+a is divisible by 28

Example:
Number N = 98 76 54 32 08 = 14 20 26 4 8 =
-12(    14 20 26 4)+8 =
-12(-12(  14 20 26+4)+8 =
-12(-12(-12( 14 20+26)+4)+8 =
-12(-12(-12(-12*14+20)+26)+4)+8 =
-12(-12(-12*    20+26)+4)+8 =
-12(-12*        10+4)+8
-12*            24+8 = 0 (mod 28)
is divisible by 28.



Code: PS, PD
P=1000, (k=8). 1008 = 36*28
Formulae:
Number N is divisible by 28 iff -8b+a is divisible by 28

Example:
Number N = 68 940 with base  P=1000
-8*68 + 940 = 396 = 14*28+4
Number N isn't divisible by 28

number 27

Code: PU  
P=100 (k=73=19 (mod 27) )

Formulae:
Number N is divisible by 27 iff 19*b+a = -8*b+a is divisible by 27

Example:
Number N = 98 76 54 32 19 = 17 22 0 5 19 =
-8(  17 22 0 5)+19 =
-8(-8( 17 22 0+5)+19 =
-8(-8(-8(17 22+0)+5)+19 =
-8(-8(-8(-8*17+22)+0)+5)+19 =
-8(-8(-8*   21+0)+5)+19 =
-8(-8*      21+5)+19 =
-8*         26+19 = 0 (mod 27)
is divisible by 27.



Code: SF  19*10-7*27=1
19(10b+a) = b+19a = b-8a (mod 27)
Formulae:
Number N is divisible by 27 iff b+19a = b-8a is divisible by 27

Example:
Number N = 68931
6893-8*1 = 6885
 688-8*5 = 648
  64-8*8 = 0
is divisible by 27



Code: SF-  8*10-3*27=-1
8(10b+a) = -b+8a = -(b-8a) (mod 27)
It is the same as above 

number 26

Code: PU   
P=100 (k=74=22 (mod 26) )
Formulae:
Number N is divisible by 26 iff 22*b+a = -4b+a is divisible by 26

Example:
Number N = 98 76 54 32 22 = 20 24 2 6 22 =
-4(  20 24 2 6)+22 =
-4(-4( 20 24 2+6)+22 =
-4(-4(-4(20 24+2)+6)+22 =
-4(-4(-4(-4*20+24)+2)+6)+22 =
-4(-4(-4* (-4)+2)+6)+22 =
-4(-4*      18+6)+22 =
-4*         12+22 = 0 (mod 26)
is divisible by 26.



Code: CR PD 
P=25 (k=1)
Formulae:
Number N is divisible by 26 iff alternate sum of digits is divisible by 26

Example:
Number N = 68928 in decimal
Conversion to base P=25 is in previous post number 25
number is 4 10 7 3 with base 5^2 = 25
Formulae is: -4+10-7+3 = 2
so number N isn't divisible by 26

number 25

Code: SN
100 = 4*25
Formulae:
Number N is divisible by 25 iff least significant two digits are 00, 25, 50 or 75.

Example:
Number N = 59 60 05 44 76 75
is divisible by 25




Code: PU   
P=100 (k=75=0 (mod 25) )
Formulae:
Number N is divisible by 25 iff least significant digit is divisible by 24

Example:
Number N = 98 76 54 32 00
is divisible by 25.



Code: CR 
P=25 (k=1)
Formulae:
Number N is divisible by 25 iff least significant digit is 0

Example:
Conversion to base P=25 uses method, eg:
1) from decimal system P=10: P=10*1/2, next P^2 into 25
2) P=100*1/4
Number N = 68928 with base P=10
Conversion from P=100*1/4:  digits in bracket [] multiply by 4
[6 89 | 28] =  [24 | 356] 28 = [24 | 14*25+6] 25+3
[24] = 96 = 3*25+21
repair of digits, they are not bigger than 25: 
96 356 28 = 3 (21+14) (6+1) 3 = 4 10 7 3
or
from P=10*1/2:
[6 8 9 2 | 8] = [12 16 18 4] 8
[1 3 7 8 | 4] = [2 6 14 16 ] 4
[2 7 5 | 6] = [4 14 10] 6
[5 5 | 0] = [10 10] 0
[1 1 | 0] = [2 2] 0
[2 | 2] = 4 2
number is  4 2 0 0 6 4 8 = 4201203 with base P=5
number is 4 20 12 03 = 4 10 7 3 with base 5^2 = 25
Number N isn't divisible by 25 because remainder is 3