Osculators

In this post I'm describing a very interesting method of determining the divisors of any given number that I came across in a post by Sohel Sahoo:

All credit is given to Sohel Sahoo for his method and all I've done in this post is to rephrase his method so that it is easier for me to understand. 

The assertion is made that there exist two specific numbers (called osculators: one positive, the other negative) for any divisor. The divisor's positive or the negative osculator can be used to determine if a given number has that particular divisor. The sum of the absolute values of the osculators is equal to the divisor.

Procedures to find the osculators of a given divisor

(7 and 13 will be used as examples):

1. If the divisor is a single digit, work with the smallest multiple that has two digits.
   
   In the case of 7, the smallest two digit multiple is 14 so we work with that.
   
   In the case of 13, there is no need to modify it.
   
   
2. Let the osculator be \(x\) and multiply the unit digit of the divisor by \(x\).
   
   In the case of 14, this gives \(4x\).
   
   In the case of 13 this gives \(3x\).
   
   
3. Add the result obtained in step 2 to the remaining digits to obtain an expression in \(x\).
   
   In the case of 14 this gives the expression \(1+4x\).
   
   In the case of 13 this gives the expression \(1+3x\).
   
   
4. Find the smallest positive and negative values of \(x\) that make the expression divisible by the divisor. These are the positive and negative osculators for the divisor.
   
   In the case of 7, \(1+4 \times 5=21\) and \(1+4 \times -2 =-7\) and so the osculators are 5 and -2. Note that |5|+|-2|=7.
   
   In the case of 13, \(1 + 3 \times 4=13\) and \(1+ 3 \times -9 = -26\) and so the osculators are 4 and -9. Note that |4|+|-9|=13.

Doing this for other divisors generates the table shown below:

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

Some tips for remembering the osculators:

Any divisor 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:

\(a9=10a+9=9a+a+9=9a+9+a=9(a+1)+a\)

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 the unit 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.

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 and 1001 e.g. 111111, 222222 and 333333 are divisible by these numbers.

Formula for using the osculators to test divisibility:

We'll use 69125 and divisibility by 7 as an example to illustrate the use of the formula. We'll use both the positive and negative osculators (although in practice, only one is needed):

1. Firstly, multiply the osculator of the divisor by the unit digit of the number that is being tested for divisibility by that divisor.
   
   In the case of 69125 being test for divisibility by 7, this means 5 x 5 = 35 using the positive osculator of 5 and 5 x -2 = 10 using the negative osculator of -2.
   
   
2. Add the result so obtained on multiplication to the remaining of digits if using the positive osculator, subtract the result if using the negative osculator.
   
   Thus 69125 --> 6912 + 25 = 6937 or 6912 - 10 = 6902.
   
   
3. Repeat processes 1 and 2 until the number is small enough to be recognised as a multiple of the divisor or not. 
   
   Thus 6937 --> 693 + 35 = 728 --> 72 + 40 = 112 --> 11 + 10 --> 21 which is 7 x 3.
   6902 --> 690 - 4 = 686 --> 68 -12 = 56 which is 7 x 8.
   
   
4. If the result is a multiple of the divisor then the divisor is confirmed.
   
   69125 does reduce to a multiple of 7 and so 7 is a divisor. It can also be seen that using the smaller of the two osculators (ignoring signs) makes for easier calculations.

Another example:

What happens if 69125 were tested for divisibility by 13? We'll use the 4 as the osculator as it is the smaller of the two.

69125 --> 6912 - 20 = 6892 --> 689 - 8 = 681 --> 68 - 4 = 64 which is clearly not a multiple of 13 and so we conclude that 69125 is not divisible by 13.

Usefulness of Divisibility Tests

One can argue that learning divisibility tests like this is pointless in this technological age but I think such mental activities combat mental decline (I am a septuagenarian so that's important) and reduce our reliance on technology by making us more confident and self-sufficient (as a counterbalance to the encroachments of AI).

Note on Osculators and Osculation

It should be noted that the terms osculator and osculation have special meanings in Vedic Mathematics. An osculator is an algorithm for performing osculation while osculation means the determination of whether a number is divisible by another by means of certain operations on its digits. The meaning of osculation is thus different from that of mainstream mathematics where the term means a contact between curves or surfaces, at which point they have a common tangent. Thus we can speak of osculating circles, meaning two circles that touch at a point through which a common tangent passes. Here is a link to more information on Vedic Mathematics.