6174

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:

  1. Take any four-digit number, using at least two different digits (leading zeros are allowed).
  2. Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary.
  3. Subtract the smaller number from the bigger number.
  4. 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]

← 6173 6174 6175 →
Cardinalsix thousand one hundred seventy-four
Ordinal6174th
(six thousand one hundred seventy-fourth)
Factorization2 × 32 × 73
Divisors1, 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 numeralVMCLXXIV, or VICLXXIV
Binary11000000111102
Ternary221102003
Senary443306
Octal140368
Duodecimal36A612
Hexadecimal181E16

Other "Kaprekar's constants"
Main article: Kaprekar's routine § Definition and properties

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:
    • 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
  1. Nishiyama, Yutaka (March 2006). "Mysterious number 6174". Plus Magazine.
  2. Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica. 15: 244–245.
  3. Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics. 13 (2): 81–82.
  4. Hanover 2017, p. 1, Overview.
  5. "Kaprekar's Iterations and Numbers". www.cut-the-knot.org. Retrieved 2022-09-21.
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