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The God of small things!
by Quant Ghost - Monday, 24 November 2008, 10:36 PM
 

Quant Ghost

I hope the readers did not find the first article difficult to digest. The readers are advised not to hurry themselves. They are to go through these thoughts with absolute peace of mind, for the mind shall reveal its true potential one day for a good cause. No more lectures. Let us get back to mathematics. Have you noticed how a small seed holds the key to a big tree? Here is a question to prove the point-

problem

To know the answer to this question, you will have to pay homage to the simplest of all rules- the digit-sum rule.

What is Digit Sum?

Given a number N1, all the digits of N1 are added to obtain a number N 2 . All the digits of N2 are added to obtain a number N3, and so on, till we obtain a single digit number N. This single digit number N is called the digit sum of the original number N1.

Example: What is the digit sum of 123456789?

Answer: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 --> 4 + 5 = 9. Hence, the digit sum of the number is 9.

Note: In finding the digit-Sum of a number we can ignore the digit 9 or the digits that add up to 9. For example, in finding the digit-sum of the number 246819, we can ignore the digits 2, 6, 1, and 9. Hence, the digit-sum of 246819 is = 4 + 8 = 12 = 1 + 2 = 3.

Digit-Sum Rule of Multiplication: The digit-sum of the product of two numbers is equal to the digit sum of the product of the digit sums of the two numbers!

Example: The product of 129 and 35 is 4515.

Digit sum of 129 = 3 and digit sum 35 = 8

Product of the digit sums = 3 × 8 = 24 --> Digit-sum = 6.

Digit-sum of 4515 is = 4 + 5 + 1 + 5 = 15 = 1 + 5 = 6.

Digit-sum of the product of the digit sums = digit sum of 24 = 6

--> Digit sum of the product (4515) = Digit-sum of the product of the digit sums (24) = 6

Applications of Digit-Sum

  1. Rapid checking of calculations while multiplying numbers

Suppose a student is trying to find the product 316 × 234 × 356, and he obtains the number 26525064.

A quick check will show that the digit-sum of the product is 3. The digit-sums of the individual numbers (316, 234 and 356) are 1, 9, and 5. The digit-sum of the product of the digit sum is 1 × 9 × 5 = 45 = 4 + 5 = 9.

--> the digit-sum of the product of the digit-sums (9) is NOT equal to digit-sum of the 26525064 (3)

Hence, the answer obtained by multiplication is not correct.

Note: Although the answer of multiplication will not be correct if the digit-sum of the product of the digit-sums is not equal to digit-sum of the product, but the reverse is not true i.e. the answer of multiplication may or may not be correct if the digit-sum of the product of the digit-sums is equal to digit-sum of the product

  1. Finding the sum of the digits of a number raised to a power

Example: The digits of the number (4)24 are summed up continually till a single digit number is obtained. What is that number?

Answer: 43 = 64. Digit sum of 64 is = 1.

424 = 43 × 43 × 43 ... × 43 (8 times)

Digit sums on both sides will be the same.

--> digit sum of 424 = digit sum of 1 × 1 × 1 × 1... (8 times) = 1

image

Example: Find the sum of the sum of the sum of the digits of 25!

25! = 1 × 2 × 3 × ... × 24 × 25. As one of the multiplicands is 9, the digit sum will be 9.

3. Determining if a number is a perfect square or not

image

It can be seen from the table that the digit-sum of the numbers which are perfect squares will always be 1, 4, 9, or 7.

Note: A number will NOT be a perfect square if its digit-sum is NOT 1, 4, 7, or 9, but it may or may not be a perfect square if its digit-sum is 1, 4, 7, or 9.

Example: Is the number 323321 a perfect square?

Answer: the digit-sum of the number 323321 is 5. Hence, the number cannot be a perfect square.

Example: A 10-digit number N has among its digits one 1, two 2s, three 3s, and four 4s. Is N be a perfect square?

Answer: We can see that the digit sum of a perfect square is always 1, 4, 7, or 9. As the digit sum of the number is 3, it cannot be a perfect square.

Now can you answer the question that I posed? I bet that you can.


 Suppose, the seed of any positive integer n is defined as follows:

                    Seed(n) = n, if n < 10
                                = seed(s(n)), otherwise,

Where s(n) indicates the sum of digits of n. For example,
Seed(7) = 7, seed(248) = seed(2 + 4 + 8) = seed(14) = seed(1 + 4) = seed(5) etc.

How many positive integers n, such that n < 500, will have seed(n) = 9? (CAT 2008)
(1) 39                           (2) 72                          (3) 81                          (4) 108                   (5) 55

We are done here for today. I hope the reader found this simple concept a useful weapon for his mathematical armoury. It would be an honour if the ghost can help you regarding your queries and doubts. Meet you on the next page after I finish my cigar!

image

Re: The God of small things!
by sabby ag - Wednesday, 7 January 2009, 10:49 PM
  should be 55......

495 is the largest number which is multiple of 9 but less than 500.
and 495 / 9 = 55
Re: The God of small things!
by ujjawal kumar - Monday, 2 February 2009, 12:12 PM
 

what about 123456789?

Is this a perfect square or not?

the sum digit is 9.

so it should be a perfect square.

But it is not a perfect square.

Re: The God of small things!
by Total Gadha - Tuesday, 3 February 2009, 11:12 AM
 

Hi Ujjawal,

See the Note below the table please.

Total Gadha

Re: The God of small things!
by vikas sharma - Friday, 13 February 2009, 04:47 PM
 

Hi

i have recently joined total gadha guys hats off to all . Relly very helpful

Re: The God of small things!
by Ashwin A - Thursday, 28 May 2009, 01:54 AM
   kudos to Q Ghost, TG in explaining concepts in such an easy and understandable manner cool
Re: The God of small things!
by I G - Thursday, 28 May 2009, 01:19 PM
  hello . .can u give me some tips to improve quant ...as to abt NCERT books whch class books shld i refer to..
Re: The God of small things!
by I G - Thursday, 28 May 2009, 02:42 PM
  and abt TG its really an awsome site....thanks TG...keep doing this work
Re: The God of small things!
by Total Gadha - Friday, 29 May 2009, 01:07 AM
  Hi IG,

Refer class 8th, 9th and 10th books, especially for geometry.

Total Gadha
Re: The God of small things!
by I G - Friday, 29 May 2009, 01:14 PM
  thanku TG for replying...
i also wanted to ask that as there is very less time for cat so what shld be my strategy for all subj ..hw much time shld i devote daily to each subj??...n till which mnth shld i plan to complete all the concepts ??
plz do reply
Re: The God of small things!
by I G - Friday, 29 May 2009, 01:51 PM
  n sir one more thing i 4got to ask...
is vedic maths helpful for calcuations.. or any other advice u wld like to give regrdng calcultns
plz gimme some links whch wld be helpful for incrsng all sorts of calculatn speed..n if u hvnt posted yet..plz post them  in TG.com asap...
thanks
Re: The God of small things!
by Deepika Khandelwal - Wednesday, 10 June 2009, 12:06 PM
  hi..TG
give the solution for seed question,how to solve this ???
Re: The God of small things!
by Software Engineer - Thursday, 11 June 2009, 01:22 PM
  Deepika,

The article says:- If one of the multiplicand is 9, the digit sum is always 9.

The seed(n) function defined above is nothing but digit sum of n. seed(n) = 9; holds true for all the multiples of 9. The total number of multiples of 9 less than 500 = 500/9 = 55 (quotient). Hence, (5).

- SE
Re: The God of small things!
by nidhi soni - Tuesday, 16 June 2009, 01:57 PM
 

hi quant ghost!!!

i hv a question....if we need find out the product of 1.32 * 3.42 * 5.5569

thn how can we use digit sum method for such prbs?????

and one more doubt....how can we solve a equ.... if we r asked to solve

423*68 /?=1250.60

 

how can we get our ans wdout wasting time in calculations for such equation?...

Re: The God of small things!
by anubhuti monga - Tuesday, 23 June 2009, 11:55 AM
 

Hi

m new to this site.i just joined today..

n m new for CAt too. please suggest me some strategies that i ned to follow fr CAT 2009.

please suggest me the books too

thanx

Re: The God of small things!
by abin abraham - Wednesday, 24 June 2009, 08:49 PM
  gr8 Quant Ghost i just lved d article
Re: The God of small things!
by Prakash jain - Friday, 26 June 2009, 05:27 AM
  qh kindly post the solution of seed problem
Re: The God of small things!
by ashish verma - Tuesday, 7 July 2009, 04:20 AM
 

A=8888^8888, B= sum of digits of A, C= sum of digits of B, D=sum of digits of C.....in this
series there will be a point where you will get sum of digits of X=X it self. find X.

 

if we change the base then how calculate sum of digits

Re: The God of small things!
by Software Engineer - Tuesday, 7 July 2009, 06:46 AM
  Ashish,

The digit sum of a number N is notìng but the remainder when N divided by 9.

Here, we need to find the digit sum of 8888^8888 i.e. divide it by 9 and get the remainder.

8888^8888 mod 9
5^8888 mod 9
Now apply Euler phi(9)=6 so divide the power 8888 by 6 and get the remainder
5^2 mod 9
7 mod 9

Hence, X=7.

In base b the digit sum rule applies to the digit (b-1). Therefore, if base is b the divide 8888^8888 by (b-1) to get the digit sum; the remainder thus obtained is the value of X.

- SE
Re: The God of small things!
by tarun bhavnani - Thursday, 30 July 2009, 09:05 PM
  quant ghost and SE,
thnx...u both Rock!!!
Re: The God of small things!
by Kunal Gupta - Sunday, 2 August 2009, 03:06 AM
 

TG,

Digit sum=9 has anything with base 10? what abt other bases?

Also, any one has 8,9,10th NCERT e-books.. pls post them toosmile 

Re: The God of small things!
by Varun Agrawal - Wednesday, 2 September 2009, 07:27 PM
  Go to ncert site.. you will find all of these ebooks. 
Re: The God of small things!
by priyanka ramani - Thursday, 3 September 2009, 12:30 AM
  hello quant ghost......

dat was fantastic....very helpful....

Note: A number will NOT be a perfect square if its digit-sum is NOT 1, 4, 7, or 9, but it may or may not be a perfect square if its digit-sum is 1, 4, 7, or 9.

for the no's having the digit sum as 1,4,7,9 is ok...bt for the no's not having 1,4,7,9 as their digit sum bt still they are perfect square,for them can u jus temme any method 2 check them...r they perfect square or not...plzzz reply me...
Re: The God of small things!
by niteesh baranwal - Wednesday, 16 September 2009, 10:38 PM
  @: UJJAWAL

the article says that the number not ending in 1,4,7 or 9 are definitely not perfect squares...
but the numbers ending with 1,4,7,or 9 may or may not be perfect squares...
Re: The God of small things!
by Tamanash Biswas - Thursday, 24 December 2009, 11:47 PM
 

Nitesh

Its not the last number but the digit sum being talked about. Numbers ending with 2,3,7 and 8 can never be perfect squares. When it comes to digit sum, perfect squares always have digit sum either of 1,4,7,or 9 but not vice versa.smile

Re: The God of small things!
by ketan jajoo - Wednesday, 6 January 2010, 12:36 PM
 

Hi Ujjawal,

Already said by the ghost that its not necessary that the numbers with sum digit = 1, 4, 7 or 9 are perfect squares.

But any number with sum digit apart from 1, 4, 7 or 9 can never be a perfect square.

Hope I am clear with my fundassmile

 

Re: The God of small things!
by akanksha pandita - Tuesday, 12 January 2010, 06:26 PM
  perhaps answer is 55
Re: The God of small things!
by akanksha pandita - Thursday, 14 January 2010, 10:05 PM
  55
Re: The God of small things!
by ankush gupta - Thursday, 21 January 2010, 05:39 PM
 

Hi  anubhuti  monga

i can guide you if u want...........u r already a brilliant person.

 

Re: The God of small things!
by Prashant Sahni - Friday, 22 January 2010, 07:43 PM
  Man I hate this 'may or may not be' stuff. Feels like even after putting so much effort in calculating the digit sum, I'm still not sure if my multiplication is correct or not.

Anyway thanks for the great article man! Really great stuff.

Regards
Re: The God of small things!
by Prashant Sahni - Friday, 22 January 2010, 08:43 PM
  I couldn't understand that logic. Do you mean to say that any number is a multiple of its digit sum? But that is not true of (say) 22 whose digit sum is 4. Is this logic exclusive to 9 and 3? Please Explain.

Regards
Re: The God of small things!
by bhupendra jantwal - Monday, 26 April 2010, 10:48 AM
  great work !!!! keep it up.
live long sir.

cheers.
Re: The God of small things!
by deepti anand - Wednesday, 2 June 2010, 12:38 AM
  hii sir..

needless 2 say...awsum article...

sir jst hv a doubt...if d digit sum is 1,4,7or9..it may b or may nt b a perfect sq..so if for a number we gt digit sum as 1 or 4 or 7 or 9...it may b or may nt b a perfect sq...bt wat if we hv 2 ans specifically....hw can we check dat..??...
Re: The God of small things!
by gaurav midha - Friday, 10 February 2012, 08:53 AM
  Wonderful article..thank you.
Re: The God of small things!
by sachin bhattad - Friday, 15 June 2012, 08:55 PM
 

thank u TG....

 

Re: The God of small things!
by Ashwani Kushwaha - Saturday, 16 June 2012, 01:57 PM
  I'm new on TG and I have a question for Sr. n guys here. I'm so weak in Quant. how can I improve my mathematics.
plz help..
Re: The God of small things!
by Manoj Rawat - Wednesday, 13 May 2015, 03:34 PM
  Great concept...Thnk you