# 6174 (number)

__: Natural number__

**Short description**
| ||||
---|---|---|---|---|

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

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

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

Divisors | 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 343, 441, 686, 882, 1029, 2058, 3087, 6174 | |||

Greek numeral | ,ϚΡΟΔ´ | |||

Roman numeral | VMCLXXIV, or VICLXXIV | |||

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} |

**6174** is known as **Kaprekar's constant**^{[1]}^{[2]}^{[3]} after the *India* n mathematician D. R. Kaprekar. This number is renowned 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 always reach its fixed point, 6174, in at most 7 iterations.^{[4]} Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 1459:

- 9541 – 1459 = 8082
- 8820 – 0288 = 8532
- 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. For numbers with three identical numbers and a fourth number that is one number higher or lower (such as 2111), it is essential to treat 3-digit numbers with a leading zero; for example: 2111 – 1112 = 0999; 9990 – 999 = 8991; 9981 – 1899 = 8082; 8820 – 288 = 8532; 8532 – 2358 = 6174.^{[5]}

## Other "Kaprekar's constants"

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".

## Other properties

- 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:
- 18
^{3}+ 18^{2}+ 18^{1}= 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18.

- 18
- The sum of squares of the prime factors of 6174 is a square:
- 2
^{2}+ 3^{2}+ 3^{2}+ 7^{2}+ 7^{2}+ 7^{2}= 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13^{2}

- 2

## References

- ↑ Nishiyama, Yutaka (March 2006). "Mysterious number 6174". http://plus.maths.org/issue38/features/nishiyama/index.html.
- ↑ 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. - ↑ Hanover 2017, p. 1, Overview.
- ↑ "Kaprekar's Iterations and Numbers". https://www.cut-the-knot.org/Curriculum/Arithmetic/Kaprekar.shtml.

## External links

- Bowley, Roger. "6174 is Kaprekar's Constant".
*Numberphile*. University of Nottingham: Brady Haran. https://www.youtube.com/watch?v=d8TRcZklX_Q. - Sample (Perl) code to walk any four-digit number to Kaprekar's Constant
- Sample (Python) code to walk any four-digit number to Kaprekar's Constant
- Sample (C) code to walk the first 10000 numbers and their steps to Kaprekar's Constant