Special Divisibility rules | one shortcut method for all numbers including primes.

Sohel Sahoo
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Sohel Sahoo: Hello guys,
           In this article you would come to know an amazing formula to check divisibility rule of any number that could help you a lot to solve problems in hands down in exams.

Rules for applying the formula:-

 There exists two specific no.s (called osculators,viz:-one is positive & other being negative) for each number.

1. Firstly, multiply the osculator to the unit digit of the given number, Which is to be examined whether it is divisible by that divisor no. Or not.

2. Then add the result so obtained on multiplication from the rest of the digits gained by cancelling that unit digit of the given number.

3. Now,if the result of this sum is that very prime no. Or a multiple of it then the given number is said to be divisible by the divisor number.

We can carry on this process repeatedly until we reach a comparatively small no. Which gives us the necessary clue as to whether the given no. is divisible by it or not.

Note:-  In case of positive osculator we can directly follow the above said procedure while for negative osculator the exception is  addition with it becomes subtraction from rest digits being multiplied its modulus (+ve) value.

Procedures to find the osculators:-

  1.    Let the osculator be 'x'.
   Hence, according to rules cited above multiply it to last digit of given number.Express it as the sum of the result of multiplication and rest digits.

2. Now check for which value of 'x'  the expression is divisible by the divisor number.

You can check from 1,2,3... So on for positive osculator manually and -1,-2,-3 ...so on for negative osculator.

Note:-   The divisor number should be of two digits if not then express it to the nearer multiple of two digits as in case of one digits its previous digits are zero which is of no use !

Relationship between two osculators:-

   Let the given no.which is to be tested be represented as  ab'  & osculator be ' x'.
                 So,according to formula we have bx+a, divisible by ab.
    Now,    bx+a—ab2  is divisible by ab
=>a—ab2+bx is divisible by ab
=>a—b(ab--x) is divisible by ab

  Clearly,(ab--x) is the negative osculator.
NOTE:- (i) Sum of both osculators (positive&negative) is equal to divisor.
                                                                                         [Excluding the sign (+)/(--)]
 (ii) By getting one we can obtain another just by placing opposite sign to difference of that osculator from divisor.
(iii)For easy & efficacy take that osculator which is small irrespectve of signs.
e.g.-
# Find the osculator for 13.
Let the osculator be 'x'.
According to rules, the required expression will be '3x+1'.
Now  checking the values of x for which the 3x+1 is divisible by 13 we find
   3×4+1=13; divisible by 13.
Hence, the osculator will be '4' for 13.
Also,3×(-9)+1=-26; divisible by 13.
Negative osculator for 13 is '-9' .

# Similarly,if we try to find osculator for 7 (being single digit)then primarily we have to convert it to nearer multiple i.e.14.
 Now, Check value of'x'for which 4x+1 is divisible by 7.
 We find,4×5+1=21; divisible by 7
Hence, osculator for 7 is '5'.
Also 4×(-2)+1=-7; divisible by 7.
Hence, negative osculator for 7 is '-2'.

 Likewise, you can test osculators for any divisor number which is generalized.

Shortcut pro* tips for finding osculators:-

Any divisor no. ending in 9 can generally be written as   'a9'.
Let the osculator be 'x'.
According to rules the expression will be      '9x+a'.
  In decimal representation we have;
       a9=10a+9
           =9a+a+9
           =9a+9+a
           =9(a+1)+a
          ~9x+a
Obviously, the positive osculator is (a+1).
 # Henceforth, owing to this the positive osculator for divisor numbers ending in 9 is just one more than its previous digits.

1. For 9,19,29,39 etc.(all ending in 9);the positive osculators are 1,2,3,4 etc.

2. For 3,13,23,33 etc. (all ending in 3) multiply them by 3 in order to get 9 in 1's place as 9,39,69,99 etc.Thus you get 1,4,7,10 etc.as positive osculators.

3. For 7,17,27,37 etc.(all ending in 7) multiply them by 7 so as to attain 9 in unit place like 49,119,189,259 etc. Hence,you get 5,12,19,26 etc.as positive osculators.

4. For 1,11,21,31 etc. (all ending in 1) multiply them by 9 thereof attain 9 as last digit as 9,99,189,279 etc. So, you get positive osculators viz. 1,10,19,28 etc.

  •    For the sake of your  understanding I have given osculators for a set of no.s  to visualise &feel comfortable in this concept as reference.  

              Number  
Positive
Osculator
 Negative
 Osculator
3
1
-2
7
5
-2
9
1
-8
11
10
-1
13
4
-9
17
12
-5
19
2
-17
21
19
-2
23
7
-16
27
19
-8
29
3
-26
31
28
-3
33
10
-23
37
26
-11
39
4
-35
41
37
-4
43
13
-30
47
33
-14
49
5
-44
51
46
-5
53
16
-37
57
40
-17
59
6
-53
61
55
-6
63
19
-44
67
47
-20
69
7
-62
71
64
-7
73
22
-51
77
54
-23
79
8
-71
81
73
-8
83
25
-58
87
61
-26
89
9
-80

  1. Test whether 416 is divisible by 13 or not.                                    

      Sol:- The suitable osculator for 13 is 4.                                   
   So, as cited above check 6 x 4+41=65 if divisible by 13
           Clearly it do so , that's why 416 is also divisible by 13.

  2.Test whether 69125 is divisible by 7 or not.
    Sol:- For 7 suitable osculator will be -2.
    So, check 6912--(5 x 2)=6902 if divisible by 7.
          Proceeding the said process repeatedly then check 690--(2x2)=686 if divisible by 7.
        Then,68--(2x6)=56 is divisible by 7 .So, the no. 69125 is divisible by 7 aswell.
     You can check this process by taking 5 also as follows.
    

Here 21 is divisible by 7;that's why 69125 is also divisible by 7.

Special Divisibility Rule:-

If any number is made by repeating a digit 6 times then the no. will be divisible by 3,7,11,13,21,37,77,91,143&1001.

e.g.111111 or 222222 or 333333 is divisible by these no.s.
Make sure if the digit is repeating 31 times say ,then upto 30 digits the no will be divisible&on dividing by these no.s remainder will be the rest one digit.

If you want to learn reason behind this comment down for full explanation & several questions.

     Hope this will serve my purpose of writing to reveal the logic &concepts behind each word of mathematics&at the end you shall be affectionated with it.Feel free to contanct us for getting answer of any query.
      Stay connected & feel the way of learning...

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6 Comments
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  1. Great stuff! However, just to point out a typo: 229 should be 119 below:

    3. For 7,17, 27, 37 etc.(all ending in 7) multiply them by 7 so as to attain 9 in unit place like 49, 229, 189, 259 etc. Hence,you get 5,12,19, 26 etc.as positive osculators.

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  2. I was impressed with your divisibility methods and made a post of my own describing it in my own words but giving you full credit: https://voodooguru23.blogspot.com/2020/11/osculators.html

    ReplyDelete
    Replies
    1. Thank you sir, it's being nice that you put your efforts to make people understand in hands down. Really appreciated for your work & my pleasure that you have spent your time to bring to my notice the printing mistake. Your other projects are quite interesting & I love to read them as a keen reader having same interest in them of math & astronomy like anything.
      Hope we shall be connected...

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    2. Yes indeed. I'll look forward to reading more of your posts. Talking of astronomy, you probably know that there's a conjunction of Jupiter and Saturn coming up on December 21st (although it will be just after midnight on the 22nd here in Jakarta). The celestial latitudes are almost identical so it should be quite a sight to see (although views of the night sky are not so good here). Where do you live?

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  3. Dear Sahoo, Thank you very much for posting such an useful site for the students who are preparing for different competitive examinations. I will be using your blog post for students of Tamilnadu by translating it into Tamil. Your kind contribution will help large number of rural students. All the best to you.

    ReplyDelete
    Replies
    1. Indeed I would love ❤️ to post more and more research papers like this of mine which I have not published yet for more than 1 & 1/2 years.
      Great to hear such lovely response 😊 and you made my day by taking it further closer towards my dream of making mathematics more interesting and practical to use in daily life. Moreover to reach out large section of people all over the globe 🌍 without any language restrictions and monetary value.

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