6174 (number)
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Cardinal | six thousand one hundred seventy-four | |||
Ordinal | 6174th (six thousand one hundred seventy-fourth) | |||
Factorization | 2 × 32 × 73 | |||
Greek numeral | ,ϚΡΟΔ´ | |||
Roman numeral | VMCLXXIV, or VICLXXIV | |||
Binary | 11000000111102 | |||
Ternary | 221102003 | |||
Octal | 140368 | |||
Duodecimal | 36A612 | |||
Hexadecimal | 181E16 |
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 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 1495:
- 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.
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 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 + 181 = 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
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.
External links[]
Wikimedia Commons has media related to 6174 (number). |
- 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
- JAVA executable : run as "java -jar NumberSystemMagic.jar". Download JAR : https://www.dropbox.com/s/wsdo8766w01rdha/NumberSystemMagic.jar?dl=0
- Arithmetic dynamics
- Mathematical constants
- Integers