|Cardinal||six thousand one hundred seventy-four|
(six thousand one hundred seventy-fourth)
|Factorization||2 × 32× 73|
- 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. 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.
Other "Kaprekar constants"
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".
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6174 is a Harshad number, since it is divisible by the sum of its digits:
6174 = (6 + 1 + 7 + 4) × 343.
6174 is a 7-smooth number, i.e. none of its prime factors are greater than 7.
6174 is a practical number, because an arbitrary number less than 6174 can be represented as a sum of various factors of the number 6174. This is not an uncommon property, and the nearest neighboring practical numbers are 6160, 6162, 6180, 6188.
6174 can be written as the sum of the first three degrees of the natural number 18:
18³ + 18² + 18 = 5832 + 324 + 18 = 6174.
The sum of squares of the prime factors of 6174 is an exact square:
2² + 3² + 3² + 7² + 7² + 7² = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13².
|Wikimedia Commons has media related to Kaprekar's constant.|
- 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