## Kaprekar Number 6174

Few days back I posted an article based on the interesting properties of 153. Lot of people got very excited to know about similar other numbers with such interesting properties. Today I will be discussing about another such interesting number: 6174

### Kaprekar’s Constant

6174 is known as Kaprekar’s constant named after the Indian recreational mathematician D. R. Kaprekar. I’ve written about  D.R.Kaprekar and contribution of other Indian mathematicians. 6174 has got a very interesting property. To know what that mysterious property is take any four-digit number. Arrange the digits in ascending and then in descending order to get two four-digit numbers. Then subtract the bigger number from the smaller number. If we keep on repeating this process we will end up in 6174. This process is called Kaprekar’s routine. All the numbers will yield 6174 in 7 or less than 7 iterations.

Example

Let’s randomly choose any number, say 4518:

Now, arranging the digits in ascending and then in descending order to get two four-digit numbers.

8541-1458 = 7083

8730-0378 = 8352

8532-2358 = 6174

Hence we get 6174 in 3 iterations.

4651 reaches 6174 after 7 iterations

6541-1456 = 5085

8550-558 = 7992

9972-2799 = 7173

7731-1377 = 6354

6543-3456 = 3087

8730-378 = 8352

8532-2358 = 6174

Try it for any 4-digit number yourself and see if it works.

Questions

1. For a specific set of numbers Kaprekar’s routine will not work. Can you tell me what numbers will those be?
2. If you follow Kaprekar routine with any 3 digit number it will also result in one specific number. Can you find out that 3 digit equivalent constant?
3. The result of each iteration of Kaprekar’s routine is a multiple of 9. Can you explain why?

Hint: you have seen the application of similar mathematical logic in the earlier post – mind reading trick.

## Interesting Properties of 153

Mathematics is something beautiful if you can see how interesting each natural number is. All numbers are interesting, but some numbers are more interesting than others.

In my earlier posts I have discussed about such interesting numbers –

One such very interesting number is 153. I figured out few properties of 153 myself and felt proud on my observations. However when I researched more I was embarrassed at the paucity of my observations.  I swear now that each number is a study in itself.

I am listing below some attention-grabbing and curious property of 153. For better understanding, I have linked certain terms to wikipedia-

## Is 0.999…= 1?

Many a times we have made 0.999….= 1. But we always thought it’s an approximation, they are not equal though.

It might be surprising for many of us to know that 0.999….. is actually EQUAL to the integer 1. It can be proved like this,

If x = 0.999…, then 10*x = 9.999… so by subtracting the first equation from the second, we get

9*x = 9.000…

## How to convert from decimal to other number systems

This post will be of special interest for people who are regularly in touch with mathematics. Students preparing for competitive examinations usually have Base System (Number Systems) in the list of their topics under quantitative aptitude.

Conversion from decimal to binary and other number bases

In order to convert a decimal number into its representation in a different number base, we have to be able to express the number in terms of powers of the other base. For example, if we wish to convert the decimal number 100 to base 4, we must figure out how to express 100 as the sum of powers of 4.

100 = (1 x 64) + (2 x 16) + (1 x 4) + (0 x 1)

= (1 x 4^3) + (2 x 4^2) + (1 x 4^1) + (0 x 4^0)

Then we use the coefficients of the powers of 4 to form the number as represented in base 4: Read More

## Munchausen Number

Munchausen Number is a number that is equal to the sum of its digits each raised to a power equal to the digit.

Munchausen number is also called perfect digit-to-digit invariant (PDDI) because of the above feature.

The only Munchausen numbers are 1 and 3435.

Specialty of  3435 – Read More

## What are Palindromes?

Palindromes are very special kind of numbers. Typically a palindrome can be described as a number, word, sentence, etc. which reads same forward and backward. Specifically with regards to numbers, Palindromes are numbers which are symmetrical, i.e. they remain the same even when their digits are reversed.

For example 14641 is a Palindrome. In fact all the single digit numbers and numbers with same digit repeated are palindromes. So all numbers like 1,2,3…8,9,11,22,99,111,etc. are palindromes.

## Properties of Palindrome Numbers

### Property #1

Reverse a non-palindromic number and add it to the original number. We will get a palindromic number by repeating this process. We may even get a palindromic number in first go. For example, let the original number be 37 (non-palindromic). Add reverse of it 73 to 37, we get 110 (not a palindromic number). Therefore repeat the process. 110 + 011 = 121 (palindromic number). Another example, 16+61 = 77 (palindromic number).

Any number that never becomes palindromic in this way is known as Lychrel Number. The most famous Lychrel number is 196. Check out the calculations for yourself!

### Property #2

A palindromic number in one base may or may not be palindromic in any other base.  For example, 1991 is palindromic in both decimal and hexadecimal (7C7)

### Property #3

Certain powers of palindromes made up of digit 1,2 and at times 3 are mostly palindromes.

For example,

• 11^2 = 121
• 22^2 = 484
• 101*101=10201
• 111*111=12321
• 121*121=14641
• 202*202=40804
• 212*212=44944

There are, however, an infinite number of cases as demonstrated here:

• 11^2 = 121, 101^2 = 10201, 1001^2 = 1002001, 10001^2 = 100020001, etc.
• 22^2 = 484, 202^2 = 40804, 2002^2 = 4008004, 20002^2 = 400080004, etc.

### Property #4

All even digit palindromes are divisible by 11. There are many prime palindrome numbers also like 101, 131, 151, 181, and 191

Similar to palindromic numbers, 1089 and 6174 (Kaprekar Constant) have beautiful properties

## Palindrome Challenge for You

Based on the above knowledge take this very interesting palindrome challenge –

1. Give 2 examples of known “Lychrel Number”, other than 196.
2. Give me the most recent palindromic date.
3. Most of us have lived through two palindrome years, 1991 being the last one. Only 11 years separate 1991 and 2002. Most palindrome years are separated by 110 years.Has there ever been a time when two palindrome years have been separated by less than 11 years?  (Here I am not talking about single digit palindromes)

There are many other interesting features related to palindromes.  Share some special feature which you could figure out.

## Divisibility Rules for 7 , 11 and 13

A guest post by Dr. Cecily Zacharias from Oklahoma city, The United Sates of America. Currently she is an instructor at Oklahoma City Community College

Rules for divisibility of 7 , 11 and 13

It is equally good for 11 and 13.

Step 1    Divide into groups of three from the right.                                                      245782          245  782

______________________________________________________________-1           1
Step 2.   Write 1,-1,1,-1(alternate 1 and -1) in a row above the number          245       782
( start at the right end  and go left)

____________________________________________________________-1            1
Step 3.   Divide each Group by 7 (or 11 or 13, whatever the divisor is )     245         782
0             5

* You can avoid step 2, by simply subtracting first remainder from the second. In this case it will be simply, 5 – 0 = 5
Step 4.  Multiply the corresponding numbers in the top row and bottom row and add     0x -1  +  5x 1   = 5

** Step 4 can also be avoided.
Step 5. a.  If the sum obtained is zero, The number is divisible by 7 (or 11 or 13 )
b. If the sum is positive, then that is the remainder when we divide the number by 7 (  or 11 or 13 )
c. If the sum  is negative, then add 7 (or 11 or 13 )  to get the remainder.
The sum is always less than the divisor.

In the example given, the sum is 5 . Which can be verified.
When 245782 is divided by 7 by long division, the quotient is 35111 and remainder is 5.

If we test for divisor 11, the bottom row will be    3      1
The sum of products of the two rows is -2. since it is negative , add 11 .
So the remainder will be 9
Actual division gives the quotient to be 22343 and remainder 9.

It is the same  method for dividing by 13 too.

If you want you can simplify the  steps 2 ,to 4 as
Find the remainders in each group and alternately add and subtract the remainders starting from  the right. Then use step 5.

On behalf of all the QuickerMaths.com users, I  am highly grateful for her contribution.

## Divisibility Rule of 7, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47

You might have seen divisibility rules of various numbers. But most of them very conveniently skip the ones which are very difficult and a divisibility rule for which is very much required.  This post includes the divisibility rule for some such numbers like 7, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.

While reading this you have to be little patient. Read this carefully and try to apply it practically. If you master divisibility rules or tests explained below, I am sure these will come very handy in various examinations including competitive ones. Read More

## Beauty of Numbers

In this post I am sharing with some very interesting numerical symmetries. I got it from varied sources but the most interesting is the book from Shakuntala Devi titled Figuring: The Joy Of Numbers. It’s a must read for all math lovers.

So here you go –

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321 Read More

## Ramanujan Number

When you love mathematics you can see magic in numbers. Your face gets lit up when you observe something new about a number. Something similar and very interesting happened with great Indian mathematician

You can see the title of this post is Ramanujan Number. You might have already guessed that he might have a stumbled up on some very interesting number with some peculiar characteristics. If you have guessed that, you are right.  Ramanujan number is 1729.

1729 is also known as the Hardy – Ramanujan number . This number is also called the Taxicab number.

Ramanujan number is so named after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to the Indian mathematician Srinivasa Ramanujan.

In Hardy’s own words:

“I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number… Read More