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
- Get the Mod by 10 of the number and multiply it by 4 – let this be n1
- Get the Div by 10 the Original Number multiply it by 3 – Let this be n2
- 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
- N1 = 5*4 = 20
- N2 = 45676*3 = 137028
- N3 = 137048
- N1 = 8*4 = 32
- N2 = 13704*3 = 41112
- N3 = 41112+32 = 41144
- N1= 4*4 = 16
- N2=4114*3 = 12342
- N3=12342+16 = 12358
- N1 = 8*4 = 32
- N2 = 1235*3 + 32 = 3737
- N3 = 3737
- N1 = 7*4 = 28
- N2 = 373*3 = 1119
- N3 = 1119+28 = 1147
- N1 = 7*4 = 28
- N2 = 114*3 = 442
- N3 = 470
- N1 = 0
- N2 = 47*3 = 141
- N3 = 141
- N1 = 4
- N2 = 14*3 = 42
- N3 = 46
How to find if a number is Divisible by 31
- 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
- Get the DIV of the number with base 10 = let this number be n2
- New Number = n1 – n2
- 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
- N1 = 3 * 3 = 9
- N2 = 76145
- New Number = N2 – N1 = 76145-9 = 76136
- N1 = 6 * 3 = 18
- N2 = 7613
- New Number = 7613 – 18 = 7595
- N1 = 5 * 3 = 15
- N2 = 759
- New Number = 759-15 = 744
- N1 = 4 * 3 = 12
- N2 = 74
- New Number = 74 – 12 = 62
- N1 = 2 * 3 = 6
- N2 = 6
- New Number = 6 – 6
How to find if a number is Divisible by 29
- Get the Mod of the number using 100. Multiply it by 2 – Let this be n1
- Get the Div of the Number using 100. Multiply it by 3 – Let this number be n2.
- Subtract N1 from N2 – to get the new Number
Iterate till you find a number divisible by 29
Example – 6802443 = 29 * 234567
- N1 = 43*2 = 86
- N2 = 68024*3 = 204072
- N3 = 204072-86 = 203986
- N1 = 86 * 2 = 172
- N2 = 2039*3 = 6117
- N3 = 6117-172 = 5945
- N1 = 45*2 = 90
- N2 = 59*3 = 177
- N3 = 87
N1 = 87 is already divisible by 29
How to find if a number is Divisible by 23
- N1 – Mod of the Number by 10 and multiply it by 9 – let this be n1
- N2 – Div the number by 10 and Multiply it by 2 – Let this be n2
- N3 = Abs( n1 – n2), if this is divisible by 23 then the original number is also divisible
Example – 2839488
- N1 = 8 * 9 = 72
- N2 = 283948*2 = 567896
- N3 = 567896 – 72 = 567824
- N1 = 4*9 = 36
- N2= 56782*2 = 113564
- N3 = 113564-36 = 113528
- N1 = 8*9 = 72
- N2 = 11352*2 = 22704
- N3 = 22704 -72 = 22632
- N1 = 2*9 = 18
- N2 = 2263*2 = 4526
- N3 = 4526-18 = 4508
- N1 = 8*9 = 72
- N2 = 450*2=900
- N3 = 900-72 = 828
- N1 = 8*9 = 72
- N2 = 82*2 = 164
- N3 = 164-72 = 92
- N1 = 2*9 = 18
- N2 = 9*2 = 18
- N3 = 0 Hence divisible by 23
How to find if a number is Divisible by 19
- Get the Mod 10 of the Number . Get the Double of the number = n1
- Get the Div 10 of the original Number = n2
- New Number = N3 = N2 + N1
- Repeat this until u get a number that is divisible by 19 or it is not
Example – 2345664 = 19 * 123456
- N1 = 4*2 = 8
- N2 = 234566
- N3 = N2+N1 = 234566+8 = 234574
- N1 = 4*2 = 8
- N2 = 23457
- N3 = 23457+8 = 23465
- N1 = 5*2 = 10
- N2 = 2346
- N3 = 2346+10 = 2356
- N1 = 6*2 = 12
- N2 = 235
- N3 = 235+12 = 247
- N1 = 7*2 = 14
- N2 = 24
- N3 = 24+14 = 38
We know 38 is divisible by 19
How to find if a number is Divisibile by 17
- 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
- Get the DIV of the number with base 10 = Multiply this number by 2 = let this number be n2
- New Number = n2 + n1
- 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
- 5876526 Mod 10 = 6 – multiple 6 by 7 = 42 = n1
- 5876526 DIV 10 = 587652 = 587652 * 2 = 1175304 = n2
- New Number = 1175304 + 42 = 1175346
- 1175346 Mod 10 = 6 – multiple 6 by 7 = 42 = n1
- 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
- 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
- Get the DIV of the number with base 10 = let this number be n2
- New Number = n2 – n1
- 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
- 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
- Get the DIV of the number with base 10 = let this number be n2
- 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
- 8638 Mod 10 = 8 – multiple 8 by 2 = 16 = n1
- 8638 DIV 10 = 863 = n2
- New Number = 863 – 16 = 847
Repeat
- 847 Mod 10 = 7 – multiply by 2 = 14 = n1
- 847 Div 10 = 84 = n2
- New Number = 84 – 14 = 70
Repeat
- 70 Mod 10 = 0
- 70 Div 10 = 7
- New Number = 7 — Hence 8638 – original number is divisible by 7