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Finding divisibility through seed numbers
by Total Gadha - Wednesday, 17 December 2008, 08:09 PM
 

Seed Numbers

Seed Numbers are used to find if a given number is divisible by a prime number. Although the concept is not used often, for some number, the divisibility rules cannot be applied and seed numbers come in handy there.

Every odd number (consider only odd prime numbers) gives unit digits of 1 and 9 in two of their first 10 multiples. For example, 3 × 3 = 9, and 3 × 7 = 21. For 17, 17 × 3 = 51, and 17 × 7 = 119. You can do this for any odd prime number and see that it is true. Now numbers ending in 1 or 9 can be written as multiples of 10 ± 1. For example 51 = 5 × 10 + 1, 119 = 12 × 10 - 1, etc. Now these numbers which are multipled by 10 (5 and 12 in this case) are the seed numbers for a particular prime number. The numbers are taken negative in the case of + 1 and positive in the case of -1. Therefore, every prime number has two seed numbers. In the above example, the seed numbers of 17 are -5 and 12. Here are seed numbers of some prime numbers listed down.

Numbers

Multiples ending in 1

Multiples ending in 9

Seed numbers

3

21 = 2 × 10 + 1

9 = 1 × 10 - 1

- 2, 1

7

21 = 2 × 10 + 1

49 = 5 × 10 - 1

- 2, 5

13

91 = 9 × 10 + 1

39 = 4 × 10 - 1

- 9, 4

17

51 = 5 × 10 + 1

119 = 12 × 10 - 1

- 5, 12

19

171 = 17 × 10 + 1

19 = 2 × 10 - 1

- 17, 2

23

161 = 16 × 10 + 1

69 = 7 × 10 - 1

- 16, 7

How to use seed numbers:

Suppose you want to find out if 9044 is divisible by 17. You know that the seed number of 17 is -5. Here is what you do:

  • Take out the unit digit of the number, multiply it with the seed number and add it to the number left after removing the unit digit. Therefore, take out the unit digit of 9044 i.e. 4, multiply it by -5, 4 × -5 = -20, and add it to the number left, i.e. 904. 904 - 20 = 884.
  • Keep repeating this operation. For 884, take out 4, multiply it by -5 --> 4 × -5 = -20, add it to number left --> 88 - 20 = 68.
  • In the end you will come up with a single digit or two digit number. If this number is divisible by the prime number, the original number is divisible by the prime number. Here, 68 is divisible by 17, therefore 9044 is divisible by 17.

Let's do it once more.

Find if 43985 is divisible by 19.

We use the seed number 2 for 19.

  • First step: 4398 + 5 × 2 = 4408
  • Second step: 440 + 8 × 2 = 456
  • Third step: 45 + 6 × 2 = 57

Now 57 is divisible by 19. Therefore, 43985 is divisible by 19.

You can use seed numbers to find divisibility by any prime number.

Total Gadha

Re: Finding divisibility through seed numbers
by jaspal singh - Tuesday, 3 March 2009, 08:08 PM
  very well expalined...... but would u tell some more usage of this concept.... if their is something exist....
Re: Finding divisibility through seed numbers
by arnab _quant-um - Wednesday, 4 March 2009, 04:31 PM
  TG sir Isnt this the osculator funda....?
Re: Finding divisibility through seed numbers
by Total Gadha - Thursday, 5 March 2009, 07:35 PM
  yep.
Re: Finding divisibility through seed numbers
by Subrahmanyam Kancherla - Friday, 10 July 2009, 11:20 AM
  what is osculator  yaar....
Re: Finding divisibility through seed numbers
by Subrahmanyam Kancherla - Friday, 10 July 2009, 11:49 AM
 

Hi TG

Thanks for lucid article......1 doubt..

Is 5 is an  exception for this.Ofcourse finding remainder is easy,just want to know if there is any other exception...

Re: Finding divisibility through seed numbers
by Subrahmanyam Kancherla - Friday, 10 July 2009, 11:52 AM
 

Hi TG

Thanks for lucid article......1 doubt..

Is 5 is an  exception for this.Ofcourse finding remainder is easy,just want to know if there is any other exception...

Re: Finding divisibility through seed numbers
by iim freak - Sunday, 19 July 2009, 12:25 AM
  hi Subrahmanyam

another exception which i can think of is 2 smile
Re: Finding divisibility through seed numbers
by lipsa syal - Sunday, 19 July 2009, 07:24 AM
 

hi tg sir,

              the topic was really good. can you please also tell me the topic of factors.......smile

thanks sir.

Re: Finding divisibility through seed numbers
by Bhushan Sawarkar - Tuesday, 18 August 2009, 05:14 PM
  Hi,

Here we take -5 as a seed number and then we solve it. but what if we solve using 12 as a seed number for 17.

last two digit number comes is 64. which not divisible for 17.
Re: Finding divisibility through seed numbers
by Supriya Kumari - Tuesday, 18 August 2009, 10:12 PM
 

Hi Bhushan,

If u r talking about 9044 to be divisible by 17(wen seed no is 12),then the calculation goes like this:

9044 = 904+12*4=952=95+2*12=95+24=119 which is clearly divisible by 17.Also even if u

repeat the process after 119 u will again get 119.......

Thanks

Re: Finding divisibility through seed numbers
by pavan bagam - Wednesday, 19 August 2009, 01:01 PM
  Simply superb concept
hav a doubt...
what if there is a large power on the number 9044pow3023 like this
Re: Finding divisibility through seed numbers
by Bhushan Sawarkar - Thursday, 20 August 2009, 12:28 AM
  thanks Supriya, smile
Re: Finding divisibility through seed numbers
by saurav verma - Thursday, 20 August 2009, 01:32 AM
 

i think it wont applicable in powers

if it is applicable den plz tell me how

Re: Finding divisibility through seed numbers
by saurav verma - Thursday, 20 August 2009, 01:49 AM
 

hi subrahmanyam,

oculator simply means to bring the no near multiple of 10

suppose

if u hav to find osculator for 7

den 7*3=21 its near to multiple of 10 so we need to decrease it by 1 to make it multiple of 10.so its a so its a negative osculator and 20 is divided

by 10 its give 2 so its become -2

suppose u hav to divide

133/7

den 3*-2==6

13-6=7

it is divisible by 7

in divisibilty of 13

13*5=51 so its 1 more osculator

so we will multiply  last digit of divisor by +5

Re: Finding divisibility through seed numbers
by Pankaj Kumar - Tuesday, 13 October 2009, 10:40 AM
  Hi TG,

This is a very good conecept,thanks 4 sharing!
but again as someone asked earlier please do tell some other use of seed number apart from divisiblity test if exist.

Thanks in advance!!
Re: Finding divisibility through seed numbers
by sharad mishra - Wednesday, 6 October 2010, 02:30 PM
  nice article i must say. i have a question
that is it possible that if we keep on reducing the number and in the last we see it giving a remainder can we find something from that remiander like the number will also give the same remainder and if number is raised to certain power then we can tell remainder will be the remainder we obtained raised to the power and den divided by number.
plz throw some light.