International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 1773

ISSN 2229-5518

Test for divisibility by 7

Mr. Sangram Mohanty

Abstract- Normally, it becomes hard to check whether a large number is divisible by 7 ex. 757,344.It takes a lot of time to perform actual division and checking the result. A divisibility rule called 6-9 method is designed to verify whether a given number is divisible by 7. This method examines the digits in a number and checks whether it is divisible by 7 without performing actual division operation.

Index Terms- (a)Black numbers (b)Division (c)Divisibility

(d)Extensions of 6-9 method

(e)Irreducible numbers

(f) Integer

(g)Pohlman-Mass method for divisibility by 7 (h)Unit digit

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INTRODUCTION

6-9 method is a divisibility rule that checks whether a given number is divisible by 7 or not without performing actual division. The number N ϵ {n: n is an integer}.
A number N can be tested by following these steps as given below:-
1) The unit digit of the number was made 9 times.
2) Rest of the digits of that given number was made 6 times.
3) 9 times the unit digit of that given number was added to
6 times the rest of the digits.
4) The sum obtained was checked if it is a multiple of 7 .If it is a multiple of 7 then the original number is divisible by 7. If not, then the number is not divisible by 7.
5) Step 4 can be repeated as many number of times until you get a small number (sum obtained) to verify.

EXISTING TECHNIQUES-

Pohlman-mass method:

In this technique for checking the divisibility of number by
7 certain steps are followed like, the unit digit of a number is twiced and subtracted from the rest digits. The difference calculated is checked whether it is a multiple of 7 or not. If
multiple then original number is divisible by 7 else not.

PROCEDURE

Method description and verification

6-9 method can be verified by taking some examples and how it is satisfied.

Example -1

91 can be verified that it is divisible by 7 as given below:-
1*9+9*6
=54+9
=63
63 is a multiple of 7, hence it is verified that 91 is divisible by 7.

Example-2

448 can be verified that it is divisible by 7 as given below:-
8*9+44*6
=72+264
=336

International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 1774

ISSN 2229-5518

Applying step 5 we proceed for 336 like
6*9+33*6
=54+198
=252
Applying step 5 we proceed for 252 like
2*9+25*6
=18+150
=168
Proceeding 168 we get
8*9+16*6
=72+96
=168
168 is a multiple of 7, hence 448 is a divisible by 7.

Example-3

4879 can be verified that it is divisible by 7 as given below:-
487*6+9*9
=3003
Applying step 5 to 3003 we get
300*6+3*9
=1800+27
=1827
Applying step 5 to 1827 we get
182*6+7*9
=1155
Applying step 5 to 1155 we get
115*6+5*9
=735
Applying step 5 to 735 we get
73*6+5*9
=483
Applying step 5 to 483 we get
48*6+3*9
=315
Applying step 5 to 315 we get
31*6+5*9
=186+45
=231
Applying step 5 to 231 we get
23*6+1*9
=139+9
=147
Applying step 5 to 147 we get
14*6+7*9
=147
147 is a multiple of 7, hence 4879 is divisible by 7.

Example-4

121 can be verified that it is not divisible by 7 as given below:-
12*6+1*9
=72+9
=81
Applying step 5 to 81 we get
8*6+1*9
=48+9
=57
57 is not a multiple of 7, hence 121 is not divisible by 7.

RESULT

Extensions of 6-9 method:

International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 1775

ISSN 2229-5518

It was experimented by 6-9 method that there are certain numbers those are divisible by 7 but are not reducible to smaller numbers.

Example

189 can be verified like

9*9+18*6
=189

105 can be verified like

9*5+10*6
=105
After applying 6-9 method to such numbers it was found that these numbers remains constant and are irreducible. Such numbers those are not reducible by 6-9 method to smaller numbers are called black numbers.
Only specific numbers like 7,21,42,63,84,105,126,147,168,189 are black numbers. These are irreducible after applying 6-9 method. These numbers can be used as multiples of 7 to verify whether a given number is divisible by 7 or not.

ACKNOWLEDGMENT

I wish thanks to my parents for kindly encouraging me to publish this research paper in IJSER journal.I also thanks to my friends Amir Ansari and Akash Kumar for giving me a good support.

CONCLUSION

This 6-9 method can be used for all integer numbers to test for divisibility by 7.

REFERENCES

(a) Pohlman–Mass method of divisibility by 7 (b) Web reference:
http://en.wikipedia.org/wiki/Divisibility_rule
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