Some Divisibility Rules Of Integers

Here are some rules for finding divisibility of numbers

I have found out these rules often before going to sleep – doing the mental calculations in my head while lying on my bed gazing into the stars.

In most cases I have verified them the next day morning with some basic arithmetic and then with some programming on a selected set of numbers

How to find if a number is Divisible by 37

  1. Get the Mod by 10 of the number and multiply it by 4 – let this be n1
  2. Get the Div by 10 the Original Number multiply it by 3 – Let this be n2
  3. New Number  = n1 + n2, if this is divisible by 37 then the original number is, else continue with the above new number the above rules

Example = 456765

  1. N1 = 5*4 = 20
  2. N2 = 45676*3 = 137028
  3. N3 = 137048

 

  1. N1 = 8*4 = 32
  2. N2 = 13704*3 = 41112
  3. N3 = 41112+32 = 41144

 

  1. N1= 4*4 = 16
  2. N2=4114*3 = 12342
  3. N3=12342+16 = 12358

 

  1. N1 = 8*4 = 32
  2. N2 = 1235*3 + 32 = 3737
  3. N3 = 3737

 

  1. N1 = 7*4 = 28
  2. N2 = 373*3 = 1119
  3. N3 = 1119+28 = 1147
  4. N1 = 7*4 = 28
  5. N2 = 114*3 = 442
  6. N3 = 470
  7. N1 = 0
  8. N2 = 47*3 = 141
  9. N3 = 141
  10. N1 = 4
  11. N2 = 14*3 = 42
  12. N3 = 46

 

 

How to find if a number is Divisible by 31

  1. Get the Mod of the number with base 10 – in other words – Chop off the last digit of the number and multiply that digit by 3 = let this number be n1
  2. Get the DIV of the number with base 10 = let this number be n2
  3. New Number = n1 – n2
  4. Repeat Steps 1-3 till you either reach 31 or 0 – in which case the number is divisible by 31, otherwise it is not

Example

Take the number 761453 : which is 31 * 24563

  1. N1 = 3 * 3 = 9
  2. N2 = 76145
  3. New Number = N2 – N1 = 76145-9 = 76136

 

  1. N1 = 6 * 3 = 18
  2. N2 = 7613
  3. New Number = 7613 – 18 = 7595

 

  1. N1 = 5 * 3 = 15
  2. N2 = 759
  3. New Number = 759-15 = 744

 

  1. N1 = 4 * 3 = 12
  2. N2 = 74
  3. New Number = 74 – 12 = 62

 

  1. N1 = 2 * 3 = 6
  2. N2 = 6
  3. New Number = 6 – 6

How to find if a number is Divisible by 29

  1. Get the Mod of the number using 100. Multiply it by 2 – Let this be n1
  2. Get the Div of the Number using 100. Multiply it by 3 – Let this number be n2.
  3. Subtract N1 from N2 – to get the new Number

 

Iterate till you find a number divisible by 29

Example – 6802443 = 29 * 234567

  1. N1 = 43*2 = 86
  2. N2 = 68024*3 = 204072
  3. N3 = 204072-86 = 203986

 

  1. N1 = 86 * 2 = 172
  2. N2 = 2039*3 = 6117
  3. N3 = 6117-172 = 5945

 

  1. N1 = 45*2 = 90
  2. N2 = 59*3 = 177
  3. N3 = 87

 

N1 = 87 is already divisible by 29

 

How to find if a number is Divisible by 23

 

  1. N1 – Mod of the Number by 10 and multiply it by 9 – let this be n1
  2. N2 – Div the number by 10 and Multiply it by 2 – Let this be n2
  3. N3 = Abs( n1 – n2), if this is divisible by 23 then the original number is also divisible

 

Example – 2839488

 

  1. N1 = 8 * 9 = 72
  2. N2 = 283948*2 = 567896
  3. N3 = 567896 – 72 = 567824

 

  1. N1 = 4*9 = 36
  2. N2= 56782*2 = 113564
  3. N3 = 113564-36 = 113528

 

  1. N1 = 8*9 = 72
  2. N2 = 11352*2 = 22704
  3. N3 = 22704 -72 = 22632
  4. N1 = 2*9 = 18
  5. N2 = 2263*2 = 4526
  6. N3 = 4526-18 = 4508
  7. N1 = 8*9 = 72
  8. N2 = 450*2=900
  9. N3 = 900-72 = 828

 

  1. N1 = 8*9 = 72
  2. N2 = 82*2 = 164
  3. N3 = 164-72 = 92

 

  1. N1 = 2*9 = 18
  2. N2 = 9*2 = 18
  3. N3 = 0   Hence divisible by 23

How to find if a number is Divisible by 19

  1. Get the Mod 10 of the Number . Get the Double of the number = n1
  2. Get the Div 10 of the original Number = n2
  3. New Number = N3 = N2 + N1
  4. Repeat this until u get a number that is divisible by 19 or it is not

 

Example – 2345664 = 19 * 123456

 

  1. N1 = 4*2 = 8
  2. N2 = 234566
  3. N3 = N2+N1 = 234566+8 = 234574

 

  1. N1 = 4*2 = 8
  2. N2 = 23457
  3. N3 = 23457+8 = 23465

 

  1. N1 = 5*2 = 10
  2. N2 = 2346
  3. N3 = 2346+10 = 2356

 

  1. N1 = 6*2 = 12
  2. N2 = 235
  3. N3 = 235+12 = 247

 

  1. N1 = 7*2 = 14
  2. N2 = 24
  3. N3 = 24+14 = 38

 

We know 38 is divisible by 19

How to find if a number is Divisibile by 17

  1. Get the Mod of the number with base 10 – in other words – Chop off the last digit of the number and multiply that digit by 7 = let this number be n1
  2. Get the DIV of the number with base 10 = Multiply this number by 2 = let this number be n2
  3. New Number = n2 + n1
  4. Repeat Steps 1-3 till you either reach 17 or 34 – in which case the number is divisible by 17, otherwise it is not

Example – 5876526

  1. 5876526 Mod 10 = 6 – multiple 6 by 7 = 42 = n1
  2. 5876526 DIV 10 = 587652 = 587652 * 2 = 1175304 = n2
  3. New Number = 1175304 + 42 = 1175346

 

  1. 1175346 Mod 10 = 6 – multiple 6 by 7 = 42 = n1
  2. 1175346 DIV 10 = 117534 = 117534* 2 = 235068 = n2

New Number = 235068  + 42 = 235110

235110  Mod 10 = 0 – multiple 0 by 7 = 0 = n1

235110  DIV 10 = 23511 = 23511* 2 = 47022 = n2

New Number = 47022 + 0 = 47022

47022  Mod 10 = 2 – multiple 2 by 7 = 14 = n1

47022  DIV 10 = 4702 = 4702* 2 = 9404 = n2

New Number = 9404 + 14 = 9418

9418  Mod 10 = 8– multiple 8 by 7 = 56 = n1

9418  DIV 10 = 941 = 941* 2 = 1882 = n2

New Number = 1882 + 14 = 1938

1938  Mod 10 = 8– multiple 8 by 7 = 56 = n1

1938  DIV 10 = 193 = 193* 2 = 386 = n2

New Number = 386 + 56 = 442

442  Mod 10 = 2– multiply 2 by 7 = 14 = n1

442  DIV 10 = 44= 44* 2 = 88 = n2

New Number = 88 + 14 = 102

102 Mod 10 = 2 – multiply 2 by 7 = 14 = n1

102 Div 10 = 10 = 10*2 = 20 = n2

New Number = 34 – which is divisible by 17

How to find if a number is Divisibile by 11

This is indeed the easiest of all

  1. Get the Mod of the number with base 10 – in other words – Chop off the last digit of the number = let this number be n1
  2. Get the DIV of the number with base 10 = let this number be n2
  3. New Number = n2 – n1
  4. Repeat Steps 1-3 till you either reach 11 – in which case the number is divisible by 11, otherwise it is not

Example – 13574

13574 Mod 10 = 4 = n1

13574  DIV 10 = 1357 = n2

New Number = 1357  – 4 = 1353

 

1353 Mod 10 = 3 = n1

1353 Div 10 = 135 = n2

New Number = 135 – 3 = 132

 

132 Mod 10 = 2 = n1

132 Div 10 = 13 = n2

New Number = 13-2 = 11 – Hence the original Number 13574 is divisible

 

How to find if a number is Divisibile by 7

  1. Get the Mod of the number with base 10 – in other words – Chop off the last digit of the number and multiply that digit by 2 = let this number be n1
  2. Get the DIV of the number with base 10 = let this number be n2
  3. New Number = n2 – n1

Repeat Steps 1-3 till you either reach 7 – in which case the number is divisible by 7, otherwise it is not

Example – 8638

  1. 8638 Mod 10 = 8 – multiple 8 by 2 = 16 = n1
  2. 8638 DIV 10 = 863 = n2
  3. New Number = 863 – 16 = 847

Repeat

  1. 847 Mod 10 = 7 – multiply by 2 = 14 = n1
  2. 847 Div 10 = 84 = n2
  3. New Number = 84 – 14 = 70

Repeat

  1. 70 Mod 10 = 0
  2. 70 Div 10 = 7
  3. New Number = 7 — Hence 8638 – original number is divisible by 7

 

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