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