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]
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 propertiesedit
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:
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
January 01, 1970
6174, number, known, kaprekar, constant, after, indian, mathematician, kaprekar, this, number, renowned, following, rule, take, four, digit, number, using, least, different, digits, leading, zeros, allowed, arrange, digits, descending, then, ascending, order, . 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 6174The 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 6173 6174 6175 List of numbersIntegers 0 1k 2k 3k 4k 5k 6k 7k 8k 9k Cardinalsix thousand one hundred seventy fourOrdinal6174th six thousand one hundred seventy fourth Factorization2 32 73Divisors1 2 3 6 7 9 14 18 21 42 49 63 98 126 147 294 343 441 686 882 1029 2058 3087 6174Greek numeral ϚROD Roman numeralV MCLXXIV or VI CLXXIVBinary11000000111102Ternary221102003Senary443306Octal140368Duodecimal36A612Hexadecimal181E16 Contents 1 Other Kaprekar s constants 2 Other properties 3 References 4 External linksOther Kaprekar s constants editMain 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 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 edit6174 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 132References edit 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 edit nbsp 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 Sample C code to walk the first 10000 numbers and their steps to Kaprekar s Constant Retrieved from https en wikipedia org w index php title 6174 amp oldid 1220002120, wikipedia, wiki, book, books, library,
