Page 77 - Ganit Kaushal

3.9 PALINDROMIC NUMBERS

Palindromic numbers are the numbers which read the same from left to right and from right to left. For
example: The numbers 99, 939, 757, 3333, 12521 are palindromic numbers, because they give us the same
number whether we read them from left to right or from right to left.
The numbers 525, 151, and 555 are some examples of palindromes using the digits 34 29 48 76
1, 2 and 5. + 43 + 92 + 84 + 67
143
132
You will be surprised to know that any natural number eventually gives us a palindromic 77 121 + 231 + 341
number if we use the steps of the following Reverse and Addition procedure: 363 484
Step 1: Start with any two-digit natural number.
Step 2: Now, reverse the number and add it to the original number. (see given picture)

Step 3: Stop if you get a palindromic number, else repeat the procedure of reversing and adding.
This process will eventually lead you to a palindrome.

Readers are advised to start by taking a 3-digit natural number and use the reverse and addition procedure.
Did you finally get a palindrome?
703
186
The answer is yes (see adjoining pictures). + 307 + 681
There are numbers for which you have to repeat this procedure a large number 1010 867
of times. The number 89 gives a palindrome 8813200023188 after repeating this + 0101 + 768
1635
procedure 24 times. (Readers are advised to check it for themselves.) 1111 + 5361
Is there any 3-digit natural number that does not give a palindrome by reversing and 6996
adding?
Yes, there is. ‘196 is the only known three-digit natural number which never yields a palindrome.’

3.10 THE KAPREKAR NUMBER

The number ‘6174’ is known as the ‘Kaprekar number’ or the ‘Kaprekar constant’. It is named after the
Indian mathematician and teacher Dattatreya Ramchandra Kaprekar, who discovered this magical number.
Kaprekar received his secondary school education in Thane and studied at Fergusson College in Pune. He
attended the University of Mumbai, and received his
bachelor’s degree in 1929. For his entire career (1930– Start with a 4-digit number
1962), he was a school teacher at a government school
in Devlali, Maharashtra. Make the largest number
from these digits. Call it ‘A’
In 1949, he discovered a fascinating and magical
phenomenon when playing with four-digit numbers Make the smallest number
that do not have all digits identical. from these digits. Call it ‘B’ Use digits of C
It is explained with the help of a flow chart here. This Subtract ‘B’ from ‘A’.
is known as the “Kaprekar Procedure”.
D.R. Kaprekar Call it ‘C’. C = A – B.
(1905–1986) Follow the steps and experience the magic for yourself. Stop if C = 6174. Else

Let’s start with a four-digit number, say 6382. See the calculations below. ‘A’ denotes the largest number
formed by the digits 6, 3, 8, 2, and ‘B’ denotes the smallest number formed by these digits. ‘C’ denotes the
difference between ‘A’ and ‘B’.

Number Play 75