Bulletin of the Marathwada Mathematical Society

Vol.13, No. 1, June 2012, Pages 01-08.

A GENERAL DIVISIBILITY TEST FOR ALL

POSITIVE DIVISORS

S.M. Chauthaiwale

Department of Mathematics,

Amolakchand College, Yavatmal- 445001, M.S., India.

E-mail: shriram@chauthaiwale.com

Abstract Swami Bharati Krishna Tirthaji had explained divisibility tests for divisor ending

in 9, 3, 1 and 7 with the help of osculators and osculation methods [3]. These methods are

extended here for divisors ending in 8, 4, 2, 6 and 5 and then modified after describing

them mathematically. The osculators are recognized as solutions of specific linear

congruence relations. The osculation results obtained by three methods are compared.

Swamiji’s claim of divisibility test is generalized and proved by using properties related

to congruence relation.

1. INTRODUCTION

Swami Bharati Krishna Tirthaji elaborated on divisibility tests for

divisor ending in 9, 3, 1 and 7 in his book [3]. He explained methods of

determination of P- type and Q- type osculators and the relation between

them. Two different osculation methods are explained with the help of

numerous illustrations along with the divisibility test for said divisors. This

discussion is critically analyzed by T. S. Bhanu Murthy in his book [2].

Further Joshi S. R [5], Bemblekar C.R. [1] and Kawale G.S. [6] explained

some theoretical aspects of Swamiji’s methods in their papers. Unkalkar

V.G. explained divisibility tests for divisor ending in 8, 4, 6, and 2 in his

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book [7] but without any theoretical discussion. Extensive numerical

illustrations and modifications there in are discussed in [4].

In this article the concept of the osculator is generalised. The Osculators are

expressed as solutions of a certain linear congruence and the existence of

negative osculator is noted. An appropriate osculator for given divisor is

defined. Swamiji’s osculation methods are generalised. A new osculation

method is given and the osculation results are compared. Swamiji’s claim of

divisibility test is generalised and proved. The subject content is illustrated

at each stage by numerical examples.

2. THE OSCULATORS

In [3], the author explained methods of determining two types of osculators

Pm and Qm known as positive and negative osculator for a positive integer m

ending in 9, 3, 1 or 7. We define below two types of osculators for positive

integers ending in any non-zero digit and give the relations between them.

These definitions cover the positive and negative osculators given in [3].

The definition of an appropriate osculator for a given positive integer m is

also given.

2.1 Definitions of P- and Q- type Osculators:

(1) Pm = (k1 m + 1) / 10, where k1 = 1, 3, 9 or 7 respectively for m ending in

9, 3, 1 or 7.

(2) Qm = (k2 m – 1) / 10, where k2 = 9, 7, 1 or 3 respectively for m ending in

9, 3, 1 or 7.

(3) P'm = (k'1 m + 2) /10, where k'1= 1, 2, 4 or 3 respectively for m ending in

8, 4, 2 or 6.

(4) Q'm = (k'2 m – 2) /10, where k'2 = 4, 3, 1 or 2 respectively for m ending in

8, 4, 2 or 6.

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(5) P''m = ( m + 5 ) / 10 for m ending in 5.

(6) Q''m = ( m – 5 ) / 10 for m ending in 5.

2.2 Relations between the Osculators:

We note the following relations between the osculators which can be

verified easily.

(i) Pm < Qm for m ending in 9 and 3.

(ii) Qm < Pm for m ending in 1 and 7

(iii) P'm < Q'm for m ending in 8 and 4.

(iv) Q'm < P'm for m ending in 2 and 6

(v) Q''m < P''m for m ending in 5.

(vi) Pm + Qm = m, P'm + Q'm = m / 2, and P''m + Q''m = m / 5.

(vii) If α = Pm, P'm or P''m and β = Qm, Q'm or Q'' m then α + β = m / g, where

g = (10, m).

Remarks:

Rem 1: It can be observed that each of the P- type and Q-type osculator is a

solution of the linear congruence 10 x ≡ g (mod m).

Rem 2: If α is any P-type osculator of divisor m such that α + β = (m / g) for

some β > 0 then (– β) is also an osculator of m.

Rem 3: Swamiji and all other authors, except in book [4], used the term

negative osculator but employed β as osculator.

2.3 Appropriate Osculator for a divisor m:

Appropriate osculator Am of m is that osculator whose absolute value is

minimum among all osculators of m defined above. Thus Am = ( g + k m) /

10, where k = 1 for divisor ending in 9 or 8, k = 2 for divisor ending in 4 or

k = 3 for divisor ending in 3. And Am = ( g – k m ) / 10, where k = 1 for

divisor ending in 1, 2 or 5, k = 2 for divisor ending in 6 or k = 3 for divisor

ending in 7.

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Note: When m ends in zero, we exclude the zero while finding the osculator.

Illustrations:

(a) For m = 29, m ends in 9, g = 1, k = 1 hence Am = (1 + 29) / 10 = 3.

(b) For m = 46, m ends in 6, g = 2, k = 2 hence Am = (2 – 92) / 10 = – 9.

3. OSCULATION METHODS

Swamiji explained two osculation methods using numerical examples only

[3]. These methods are used to osculate given positive integer N by an

osculator of divisor m ending in 9, 3, 1 or 7. Later authors followed the same

line. Here these methods are not only extended for other divisors but also

modified and refined, after describing them mathematically with the help of

the osculation function ‘Os’ from Z (Set of integers) to Z. A new osculation

method which avoids the use of osculation function is suggested. The

purpose of introducing this method is to state and prove a general divisibility

test given in the last section.

3.1 Osculation Function:

For any (n+1) digit integer N = (an an-1 an-2 …………a3 a2 a1 a0) where an ----- a0

are digits of N, we define

N' = (N – a0) /10, where a0 is unit place digit of N. (3.1)

Thus for N = 2458, N' = 245. For N = 245, N' = 24 etc.

We define osculation function Os for given N and divisor m as follows

Os (N) = g N' + Am a0. (3.2)

3.2 First Method of Osculation:

For given N and divisor m, osculation results are given by the sequence of

integers b0, b1, b2, ---- bn defined as

b0 = Os (N) and br = Os (br–1), where r = 1, 2,-----, n. (3.3)

Example 3.1: Osculate N = 63936 by A74.

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Here m = 74, g = 2, A74 = 15. Hence b0 = 12876, b1 = 2644, b2 = 592,

b3 = 148, b4 = 148.

Example 3.2: Osculate N = 1094103 by A51.

Here m = 51, g = 1, A51 = –5. Hence b0 = 109395, b1 = 10914, b2 = 1071,

b3 = 102, b4 =0.

3.3 Second Method of Osculation:

For given N and divisor m, osculation results are given by the sequence of

integers b0, b1, b2, ---- bn defined as

b0 = a0, br = Os (br–1) + g r ar where r = 1, 2, -----, n. (3.4)

We select appropriate integer bc, if necessary, such that br ≡ bc (mod m),

c = 1 to n, before evaluating the next result. However use of modular

equivalent integer is optional.

Example 3.3: Osculate N = 63936 by A74.

Here m = 74, g = 2, A74 = 15. Hence from (3.4) we get b0 = 6, b1 = 96 ≡ 22

(mod 74), we select b1= 22. Now b2 = 70 ≡ – 4 (mod 74), b3 = – 36, b4 = 0.

Without using modular equivalent integers we get b0 = 6, b1 = 96, b2 = 144,

b3 = 112 and b4 = 148.

Example 3.4: Osculate N = 1094103 by A51.

Here m = 51, g = 1, A51 = – 5. Hence from (3.4) we get b0 = 3, b1 = – 15,

b2 = 25, b3 = – 19, b4 = 53, b5 = – 10, b6 = 0.

3.4 The New Method of Osculation:

For given N and divisor m, osculation results are given by the sequence of

integers b0, b1, b2, ---- bn defined as

b0 = a0, br = Am (br–1) + g r ar. (3.5)

We select appropriate integer bc, if necessary, such that br ≡ bc (mod m),

c = 1 to n, before evaluating next result. Here also use of modular equivalent

integer is optional.

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Example 3.5: Osculate N= 63936 by A74.

Here m = 74, g = 2, A74 = 15. Hence from (3.5) we get b0 = 6, b1 = 96 ≡ 22

(mod 74), b2 = 366 ≡ – 4 (mod 74), b3 = – 36, b4 = – 444 ≡ 0 (mod 74).

Without using modular equivalent integers we get b0 = 6, b1 = 96, b2 = 1476,

b3 = 22164, b4 = 332556.

Example 3.6: Osculate N = 1094103 by A51.

Here m = 51, g = 1, A51 = – 5. Hence from (3.5) we get b0 = 3, b1 = – 15,

b2 = 76 ≡ 25 (mod 51), b3 = – 121 ≡ 32 (mod 51), b4 = – 151≡ 2 (mod 51),

b5 = – 10, b6 = 51.

Without using modular equivalent integers we get b0 = 3, b1 = – 15, b2 = 76,

b3 = – 376, b4 = 1889, b5 = – 9445, b6 = 47226.

3.5 Comparison of Osculation Results:

In first method when last osculation result bn is such that bn= 0 or bn ≡ 0

(mod m) then all previous osculation results are modular equivalent. In

Example 3.1, bn = 148 ≡ 0 (mod 74) hence 63936 ≡ 12876 ≡ 2664 ≡ 592 ≡ 0

(mod 74). Similarly in Ex. 3.2, bn = 0 hence 1094103 ≡ 109395 ≡ 10914 ≡

1071 ≡ 102 ≡ 0 (mod 51).

In second and new method osculation results obtained without using

modular equivalent integers are either equal or modular equivalent to those

obtained by using modular equivalent integers. Here in Example 3.3 we

observe that first two result are equal and 144 ≡ – 4 (mod 74), 112 ≡ – 36

(mod 74), 148 ≡ 0 (mod 74). Similarly in Example 3.4 all the results are

equal and 53 ≡ 2 (mod 51). Same is true for osculation results in Example

3.5 and 3.6.

Osculation results of second method and those of new method are either

equal or modular equivalent. From Example 3.3 and 3.5 we observe that

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– 444 ≡ 0 (mod 74), 366 ≡ 70 (mod 74), 366 ≡ 144 (mod 74), 1476 ≡ 144

(mod 74), and 22164 ≡ 112 (mod 74)

The First method requires maximum time for completing the osculation

process because other two methods use modular equivalent integers for

evaluation of the next results.

For m > 100, suggested methods take more time than the time required for

finding remainder in division of N by m.

These methods are useful to test the divisibility only; but we cannot find

quotient and remainder for given N and m. This author has suggested

‘Reverse Osculation Method’ in his book [4] using the same osculator to

find quotient and remainder after testing the divisibility.

4. DIVSIBILITY TEST

Swamiji’s test of divisibility for divisor ending in 9, 3, 1 or 7 is extended

here for all other divisors ending in 8, 4, 2, 6 or 5 as follows.

4.1 The Test:

For given N and divisor m, if the last osculation result bn is such that bn ≡ 0

(mod m), then N is also divisible by m. Otherwise N is not divisible by m.

Remark 4.1: This test is valid for bn obtained by any of the osculation

methods discussed earlier.

Bhanu Murthy [2] proved Swamiji’s divisibility criterion in the form of the

theorem using congruence properties. However the statement of this

theorem, confined to divisor m with (10, m) = 1, is generalised and proved

here for any m with (10, m) = g.

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Theorem 4.1

For given positive integer N, m and appropriate osculator α of m construct

sequence of numbers b0, b1, ----, bn as follows

b0 = a0, α br–1 + g r ar ≡ br (mod m), r = 1, 2---, n. (4.1)

Then bn ≡ 0 (mod m) implies N ≡ 0 (mod m).

Proof: Here α is osculator of m hence,

10 α ≡ g (mod m) (4.2)

Multiplying both sides of (4.2) by a0 and add 10 g a1 both sides we get

10 α a0 + 10 g a1 ≡ g a0 + 10 g a1 (mod m) (4.3)

Now multiply congruence in (4.1) for r = 1, by 10 we then get

10 α a0 + 10 g a1 ≡ 10 b1 (mod m) (4.4)

Comparing (4.3) and (4.4) we get,

g a0 + 10 g a1 ≡ 10 b1 (mod m) (4.5)

Repeating similar procedure for remaining congruencies of (4.1) we get

b0 = a0, g br–1 + 10 g r ar ≡ 10 br (mod m). (4.6)

Multiplying n congruencies in (4.6) respectively by ( 10 r–1

/ g r

), r = 1 to n

we get

b0 = a0, (10 r–1

/ g r–1

) br –1 + 10 r ar ≡ (10

r / g

r ) br (mod m). (4.7)

Adding all the congruencies in (4.7) and canceling equal terms from both

sides we get

∑ 10 r ar ≡ ( 10

n / g

n) bn (mod m), r = 0 to n

Thus N ≡ ( 10n / g

n) bn (mod m) (4.8)

Now bn ≡ 0 (mod m) is given.

Hence ( 10n / g

n) bn ≡ 0 (mod m) (4.9)

Comparing (4.8) and (4.9) we get N ≡ 0 (mod m). Hence theorem is proved.

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Note: Osculation results of second method and those due to the new method

are modular equivalent hence above theorem is also applicable to former

one.

Example 4.1: Test whether 4488096 is divisible by 76 or not.

Here N = 4488096, m = 76, Am = – 15, g = (10, 76) = 2. Osculation results

as per (3.4) are b0 = 6, b1 = – 72 ≡ 4 (mod 76), b2 = – 60 ≡ 16 (mod 76),

b3 = – 24, b4 = 184 ≡ 32 (mod 76), b5 = 104 ≡ 28 (mod 76),

b6 = 140 ≡ –12 (mod 76).

Here the last Osculation result b6 is not divisible by 76 hence N is not

divisible by 76.

Note: In Example 3.1, last osculation result is divisible by 74 hence 63936 is

also divisible. Similarly in Example 3.4 last osculation result is 0 hence

1094103 is divisible by 51.

References

[1] Bemblekar C. R., A Test of Divisibility by a Number of the Form 10 k + 9, Bulletin

of the Marathwada Mathematical Society (Aurangabad), Vol.9, No.1, (June 2008), 1 - 4.

[2] Bhanu Murthy T. S., A Modern Introduction to Ancient Indian Mathematics, Wiley

Eastern Limited, (1992), 59 - 65.

[3] Shri Bharti Krishna Tirthaji, (Swamiji), Vedic Mathematics, Ed. V. S. Agrawala,

Motilal Banarsidass, (1985), 273 - 285.

[4] Chauthaiwale S. M, and Ramesh Kolluru, Enjoy Vedic Mathematics, The Art of

Living, Sri Sri Publications, Bangalore, (2010), 220 - 240.

[5] Joshi S.R, On Test of Divisibility, Mathematics education (Siwan), Vol.15, No. 2,

(1981), B1 - B2.

[6] Kawale G. S, Test for Divisibility by Prime Numbers, Bulletin of the Marathwada

Mathematical Society (Aurangabad), Vol.9, No.1, (June 2008), 55 -61.

[7] Unkalkar V. J, Excel Vedic Mathematics, Vandana Publishers, Bangalore, (2008),

85-97.