What is Kaprekar’s Constant?
D. R. Kaprekar
In 1949, an
Indian mathematician D. R. Kaprekar discovered a number 6174,
which is called Kaprekar’s constant. This number is obtained by the
following rule.
Step 1: Take any 4-digit number in which at least
two digits are different.
Step 2: Arrange the digits of the number in
the descending and ascending order to find the greatest and the smallest numbers,
adding leading zeros if required.
Step 3: Subtract the smallest number from
the greatest number to obtain a new number.
Step 4: Repeat steps 2 and 3 for the new
number obtained in step 3.
After a few
iterations, you will get the number 6174, which is known as the Kaprekar’s
constant.
What is Kaprekar’s Routine?
The above
process is known as Kaprekar’s routine which always results in a fixed
value, 6174, in at most 7 iterations.
Example
1: Find the Kaprekar’s
constant for a 4-digit number 3816.
Solution:
First
iteration:
Arranging
the digits of 3816 in descending order, we get, 8631.
Arranging
the digits of 3816 in ascending order, we get, 1368.
Their
difference = 8631 – 1368
= 7263
Second
iteration:
Arranging
the digits of 7263 in descending order, we get, 7632.
Arranging
the digits of 7263 in ascending order, we get, 2367.
Their
difference = 7632 – 2367
= 5265
Third
iteration:
Arranging
the digits of 5265 in descending order, we get, 6552.
Arranging
the digits of 5265 in ascending order, we get, 2556.
Their
difference = 6552 – 2556
= 3996
Fourth
iteration:
Arranging
the digits of 3996 in descending order, we get, 9963.
Arranging
the digits of 3996 in ascending order, we get, 3699.
Their
difference = 9963 – 3699
= 6264
Fifth
iteration:
Arranging
the digits of 6264 in descending order, we get, 6642.
Arranging
the digits of 6264 in ascending order, we get, 2466.
Their
difference = 6642 – 2466
= 4176
Sixth
iteration:
Arranging
the digits of 4176 in descending order, we get, 7641.
Arranging
the digits of 4176 in ascending order, we get, 1467.
Their
difference = 7641 – 1467
= 6174
Why is 6174 a Magical Number?
When any
4-digit number follows the Kaprekar’s routine, it reached the number 6174 in at
most 7 steps. For example, 1234 reached 6174 in 3 steps and 2014 reached 6174
in 7 steps. Therefore, 6174 is a magical number.
All the 4-digit
numbers for which Kaprekar’s routine does not reach 6174 are called repdigits.
For example, 1111, 2222, 3333, etc. are repdigits for which Kaprekar’s routine
results in 0000 in a single iteration. All other 4-digit numbers with minimum two
different digits reach 6174 if leading zeros are used to keep the number of
digits at 4.
Example
2: Show the Kaprekar’s
constant for 4333.
Solution:
Let’s follow the
Kaprekar’s routine:
4333 – 3334
= 0999 (Iteration 1)
9990 – 0999
= 8991 (Iteration 2)
9981 – 1899
= 8082 (Iteration 3)
8820 – 0288 =
8532 (Iteration 4)
8532 – 2358
= 6174 (Iteration 5)
Example
3: Verify the
Kaprekar’s constant for a number 1059.
Solution:
Let’s follow the
Kaprekar’s routine:
9510 – 0159 =
9351 (Iteration 1)
9531 – 1359
= 8172 (Iteration 2)
8721 – 1278
= 7443 (Iteration 3)
7443 – 3447 =
3996 (Iteration 4)
9963 – 3699
= 6264 (Iteration 5)
6642 – 2466 =
4176 (Iteration 6)
7641 – 1467
= 6174 (Iteration 7)
Example 4: Reach the Kaprekar’s constant using
a number 4227.
Solution:
Let’s follow the
Kaprekar’s routine:
7422 – 2247 =
5175 (Iteration 1)
7551 – 1557 =
5994 (Iteration 2)
9954 – 4599 =
5355 (Iteration 3)
5553 – 3555 =
1998 (Iteration 4)
9981 – 1899 =
8082 (Iteration 5)
8820 – 0288 =
8532 (Iteration 6)
8532 – 2358 =
6174 (Iteration 7)
Example 5: Find the Kaprekar’s constant for a
number 3456.
Solution:
Let’s follow the
Kaprekar’s routine:
6543 – 3456 =
3087 (Iteration 1)
8730 – 0378 =
8352 (Iteration 2)
8532 – 2358 =
6174 (Iteration 3)
Example 6: Verify the Kaprekar’s constant for the
number 4080.
Solution:
Let’s follow the
Kaprekar’s routine:
8400 – 0048 =
8352 (Iteration 1)
8532 – 2358 =
6174 (Iteration 2)
Example 7: Reach the Kaprekar’s constant using the
number 4609.
Solution:
Let’s follow the
Kaprekar’s routine:
9640 – 0469 =
9171 (Iteration 1)
9711 – 1179 =
8532 (Iteration 2)
8532 – 2358 =
6174 (Iteration 3)
Properties of Kaprekar’s Constant
1. 6174 = 2 × 3 × 3 × 7 × 7 × 7
Each of the
prime factors of Kaprekar’s constant (6174) is less than 7.
2. Kaprekar’s constant (6174) can be
written as the sum of the first three powers of (6 + 1 + 7 + 4), that is, 18.
We have, 181
+ 182 + 183 = 18 + 324 + 5832 = 6174
3. The sum of the squares of the prime
factors of Kaprekar’s constant (6174) is also a square number.
22
+ 32 + 32 + 72 + 72 + 72
= 4 + 9 + 9 + 49 + 49 + 49 = 169 = 132
Related Topics:
Palindromic Numbers or Palindromes
Fibonacci Series or Fibonacci Sequence
The Value of Pi