What is Kaprekar’s Constant?

What is Kaprekar’s Constant?

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

Golden Ratio

Vedic Mathematics

The Value of Pi