The number 6174 is known as Kaprekar's constant[1][2][3] after the Indian 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 digits and a fourth digit that is one 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]
| ||||
|---|---|---|---|---|
| Cardinal | six thousand one hundred seventy-four | |||
| Ordinal | 6174th (six thousand one hundred seventy-fourth) | |||
| Factorization | 2 × 32 × 73 | |||
| 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 | 11000000111102 | |||
| Ternary | 221102003 | |||
| Senary | 443306 | |||
| Octal | 140368 | |||
| Duodecimal | 36A612 | |||
| Hexadecimal | 181E16 | |||
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 "Peyush constants" named after Peyush Dixit who solved this routine as a part of his IMO 2000 (International Mathematical Olympiad, Year 2000) thesis. [6]
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 powers of 18:
- 183 + 182 + 181 = 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18.
- 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
References
- ^ Nishiyama, Yutaka (March 2006). "Mysterious number 6174". Plus Magazine.
- ^ 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". www.cut-the-knot.org. Retrieved 2022-09-21.
- ^ Hanover 2017, p. 14, Operations.
External links
- Bowley, Roger. "6174 is Kaprekar's Constant". Numberphile. University of Nottingham: Brady Haran.
- 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
This page was last edited on 21 April 2024, at 06:23