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      • 6174 (six thousand, one hundred [and] seventy-four) is the natural number following 6173 and preceding 6175.


      Kaprekar's constant


      6174 is known as Kaprekar's constant 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. Once 6174 is reached, the process will continue yielding 7641 – 1467 = 6174. For example, choose 1459:

      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.


      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".


      Applications




      = Convergence analysis

      =
      In numerical analysis, Kaprekar's constant can be used to analyze the convergence of a variety numerical methods. Numerical methods are used in engineering, various forms of calculus, coding, and many other mathematical and scientific fields.


      = Recursion theory

      =
      The properties of Kaprekar's routine allows for the study of recursive functions, ones which repeat previous values and generating sequences based on these values. Kaprekar's routine is a recursive arithmetic sequence, so it helps study the properties of recursive functions.


      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




      External links



      Bowley, Roger (5 December 2011). "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

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    Artikel Terkait "6174"

    What is the logic behind Kaprekar's Constant?

    26 Jun 2021 · Kaprekar's constant, or 6174, is a constant that arises when we take a 4-digit integer, form the largest and smallest numbers from its digits, and then subtract these two numbers. Continuing with this process of forming and subtracting, we will always arrive at the number 6174. 6174 is known as Kaprekar's constant after the Indian mathematician ...

    Mysterious number $6174$ - Mathematics Stack Exchange

    12 Agu 2012 · $6174$ is a fixed-point of this process, i.e. $7641 - 1467 = 6174$. It turns out that it is the only fixed point, and there are no nontrivial cycles. The sum of digits of the difference could also be $27$, e.g. for $6555-5556$.

    Kaprekar's constant is 6174: Proof without calculation

    Target of this post is to prove without explicitly calculating all cases that Kaprekar's routine possesses a unique fixed point which is the famous $6174$. Writing 'unique' assumes that we ignore the boring (in the sense of uninteresting) case $0000$ .

    Proof of $6174$ as the unique 4-digit Kaprekar's constant

    14 Jun 2015 · $\begingroup$ @Mythomorphic If you are still a fan of $6174$ you may take a look at a fresh answer to a Kaprekar question which is quite parallel to yours. $\endgroup$ – Hanno Commented Nov 2, 2020 at 6:27

    sequences and series - Mathematics Stack Exchange

    26 Jul 2014 · Clearly the only value for which this process is constant is 6174 but that doesn't explain why there should be convergence. One attempt at proof is to determine all the possible numbers that converge to 6174 after a single iteration, and then attempt to reason that each too can be reached by the convergence of even more numbers in such a way ...

    A strange little number - $6174$. - Mathematics Stack Exchange

    $8532 - 2358 = 6174$ What's more interesting is that with $6174$ we get. $7641 - 1467 = 6174$ and taking any four digit number we end up with 6174 after at most 7 iterations. A bit of snooping around the internet told me that this number is called the Kaprekar's constant. A three digit Kaprekar's contant is the number 495 and there's no such ...

    number theory - Is There Any Solution Of the 6174 Problem ...

    The problem is "solved" in the sense that it is easy to check (using a computer) that all 4-digit numbers except repdigits do end up at 6174. On the other hand, it doesn't seem that there is any more satisfying and principled explanation of why this process should end up at the same fixpoint, when this is not the case for 5-digit numbers.

    Is there a connection between Kaprekar’s constant ($6174$) and …

    13 Okt 2024 · Kaprekar’s constant is $6174$, and the golden ratio translated into percentages is $61.80\\%$. Pretty close. Is this explainable or just coincident?

    Using the unique factorization to find the anwer?

    Use the unique factorization theorem to write the following integers in standard factored form. a. 1,176 b. 5,733 c. 3,675 The factorization theorem states that you need to break a number into x,y...

    Why is $111111111 \\times 111111111 = 12345678987654321$

    16 Feb 2018 · I was looking around on the internet until I stumbled upon this equation. $$111111111\\times111111111 = 12345678987654321$$ How does this actually work? It is quite amazing how the number ascend and