**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, 18^{1}
+ 18^{2} + 18^{3} = 18 + 324 + 5832 = 6174

**3.** The sum of the squares of the prime
factors of Kaprekar’s constant (6174) is also a square number.

2^{2}
+ 3^{2} + 3^{2} + 7^{2} + 7^{2} + 7^{2}
= 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13^{2}

**Related Topics:**

Palindromic Numbers or Palindromes

Fibonacci Series or Fibonacci Sequence

The Value of Pi