6174 (number)

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6174
Cardinal Six thousand one hundred seventy-four
Ordinal 6174th
(Six thousand one hundred seventy-fourth)
Factorization 2 \cdot 3^2 \cdot 7^3
Roman numeral VⅯCLXXIV
Binary 1100000011110
Hexadecimal 181E

6174 is known as Kaprekar's constant or Kaprekar's operation[1][2][3] after the Indian mathematician D. R. Kaprekar. This number is notable for the following property:

  1. Take any four-digit number with at least two digits different. (Leading zeros are allowed.)
  2. Arrange the digits in ascending and then in descending 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.

The above operation will always reach 6174 in at most 7 steps and it stops there. Once 6174 is reached, the process will keep yielding 7641 – 1467 = 6174. For example, choose 5342:

5432 – 2345 = 3087
8730 – 0378 = 8352
8532 – 2358 = 6174

The only four-digit numbers for which this function does not work are repdigits such as 1111, which give the answer 0 after a single iteration. All other four-digits numbers work if leading zeros are used to keep the number of digits at 4:

2111 – 1112 = 0999
9990 – 0999 = 8991 (rather than 999 - 999 = 0)
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174

495 also has the same property for three-digit numbers.

[edit] See also

[edit] External links

[edit] References

  1. ^ Mysterious number 6174
  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.