**6174** is known as **Kaprekar's constant**^{[1]}^{[2]}^{[3]} after the Indian mathematician D. R. Kaprekar. This number is notable for the following rule:

- Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)
- Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
- Subtract the smaller number from the bigger number.
- Go back to step 2 and repeat.

The above process, known as Kaprekar's routine, will usually reach its fixed point, 6174, in at most 8 iterations.^{[4]} Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 3524:

- 5432 – 2345 = 3087
- 8730 – 0378 = 8352
- 8532 – 2358 = 6174
- 7641 – 1467 =
**6174**

The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigits such as 1111, which give the result 0000 after a single iteration. All other four-digit numbers eventually reach 6174 if leading zeros are used to keep the number of digits at 4.

| ||||
---|---|---|---|---|

Cardinal | six thousand one hundred seventy-four | |||

Ordinal | 6174th (six thousand one hundred seventy-fourth) | |||

Factorization | 2 × 3^{2}× 7^{3} | |||

Greek numeral | ,ϚΡΟΔ´ | |||

Roman numeral | VMCLXXIV | |||

Binary | 1100000011110_{2} | |||

Ternary | 22110200_{3} | |||

Quaternary | 1200132_{4} | |||

Quinary | 144144_{5} | |||

Senary | 44330_{6} | |||

Octal | 14036_{8} | |||

Duodecimal | 36A6_{12} | |||

Hexadecimal | 181E_{16} | |||

Vigesimal | F8E_{20} | |||

Base 36 | 4RI_{36} |

Note that there can be analogous fixed points for digit lengths other than four, for instance if we use 3-digit numbers then most sequences (i.e., other than repdigits such as 111) will terminate in the value 495 in at most 6 iterations. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Kaprekar constants".

6174 is a Harshad number, since it is divisible by the sum of its digits:

6174 is a 7-smooth number, i.e. none of its prime factors are greater than 7.

6174 can be written as the sum of the first three degrees of 18:

18³ + 18² + 18 = 5832 + 324 + 18 = 6174.

The sum of squares of the prime factors of 6174 is a square:

2² + 3² + 3² + 7² + 7² + 7² = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13².

**^**Yutaka Nishiyama, Mysterious number 6174**^**Kaprekar DR (1955). "An Interesting Property of the Number 6174".*Scripta Mathematica*.**15**: 244–245.**^**Kaprekar DR (1980). "On Kaprekar Numbers".*Journal of Recreational Mathematics*.**13**(2): 81–82.**^**Weisstein, Eric W. "Kaprekar Routine".*MathWorld*.

- Bowley, Rover. "6174 is Kaprekar's Constant".
*Numberphile*. University of Nottingham: Brady Haran. - Sample (Perl) code to walk any four-digit number to Kaprekar's Constant

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