Category Archives: My Creations / Inventions

An Empirical way to find divisibility

Here is an empirical way to find if a number N1 is divisible by another number N2

This can be applied to larger numbers as well. I am giving an illustrative example below with smaller numbers which can be easier to comprehend here

Say for example you want to find out if 192 is divisible by 24

Divisor is 192 and Dividend = 24

Here is the technique

Calculate for the

Divisor                  x1 = 192 mod 10 = 2                         y1 = 192 div 10 = 19

One time Calculation for the

Dividend              x2 = 24 mod 10 = 4                                           y2 = 24 div 10 = 2

Find – New Divisor = abs(x2*y1 – x1*y2) = 19*4 – 2*2 = 72

Now repeat the above steps with the new Divisor

Calculate for the

Divisor                  x1 = 72 mod 10 = 2                           y1 = 72 div 10 = 7

Find – New Divisor = abs(x2*y1 – x1*y2)  = 7*4 – 2*2 = 24

Since the new Divisor is Equal to the Dividend we infer that the Original Number 192 is divisible by 24

 

 

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