number 35

Code: PU  
P=100 (k=65=30 (mod 35) )

Formulae:
Number N is divisible by 35 iff 30*b+a = -5*b+a is divisible by 35

Example:
Number N = 98 76 54 32 15 = 28 6 19 32 15 =
-5( 28 6 19 32)+15 =
-5(-5( 28 6 19+32)+15 =
-5(-5(-5( 28 6+19)+32)+15 =
-5(-5(-5(-5*28+6)+19)+32)+15 =
-5(-5(-5*    6+19)+32)+15 =
-5(-5*      24+32)+15 =
-5*         17+15 = 0 (mod 35)
is divisible by 35.



Code: CR 
P=70=7*10

Formulae:
Number N is divisible by 35 iff least significant digit is either 0 or 35

Example:
Number N = 8512160
[8 5 1 2 1 6 | 0] = [7 14 7 42 0 14 | 20] = 7*[1 2 1 6 0 2] 20 
isn't divisible by 35

number 34

Code: PU 
P=100 (k=66=32 (mod 34) )
Formulae:
Number N is divisible by 34 iff 32*b+a = -2*b+a is divisible by 34

Example:
Number N = 98 76 54 32 10 = 30 8 20 32 10 =
-2( 30 8 20 32)+10 =
-2(-2( 30 8 20+32)+10 =
-2(-2(-2( 30 8+20)+32)+10 =
-2(-2(-2(-2*30+8)+20)+32)+10 =
-2(-2(-2*   16+20)+32)+10 =
-2(-2*      22+32)+10 =
-2*         22+10 = 0 (mod 34)
is divisible by 34.

number 33

Code: PU  
P=100 (k=67=1 (mod 33) )

Formulae:
Number N is divisible by 33 iff sum of digits is divisible by 33

Example:
Number N = 98 76 54 32 04 = 32 10 21 32 4 =
32+10+21+32+4 = 99 = 0 (mod 33)
is divisible by 33.



Code: SF  10*10-3*33=1
10(10b+a) = b+10a (mod 33)
Formulae:
Number N is divisible by 33 iff b+10a is divisible by 33

Example:
Number N = 68905
6890+10*5 = 6940
 694+10*0 =694
  69+10*4 = 109
  10+10*9 = 100 = 3*33+1
isn't divisible by 33



Code: SF-  23*10-7*33=-1
23(10b+a) = -b+23a = -(b+10a) (mod 33)
It is the same as above.

number 32

Code: PS P=100.000
b100000+a = 32*3125b+a = a
Formulae:
Number N is divisible by 32 iff least significant digit a is divisible by 32

Example:
Number N = 98476 51430 98756 
98756 = 4*24689
Number N isn't divisible by 32



Code: PU 
P=100 (k=68=4 (mod 32) )
Formulae:
Number N is divisible by 32 iff 4*b+a is divisible by 32

Example:
Number N = 98 76 54 32 00 = 2 12 22 0 0 =
4(2 12 22 0)+0 =
4(4(2 12 22+0)+0 =
4(4(4( 2 12+22)+0)+0 =
4(4(4(4*  2+12)+22)+0)+0 =
4(4(4*   20+22)+0)+0 =
4(4*      6+0)+0 =
4*       24+0 = 0 (mod 32)
is divisible by 32.



Code: CR 
P=800 (k=100*8)
Formulae:
Number N is divisible by 32 iff remainder from division by 800 is divisible by 32

Example:
Number N = 84 56 21 34
Conversion to P=100*8
[84, 56, 21 | 34] = [80, 456, 16] | 534 = 8*[10, 57, 2] | 534
The number N has least significant digit 534 = 2*267, so N isn't divisible by 32

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.