6174 (number)

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6173 6174 6175
Cardinal six thousand one hundred seventy-four
Ordinal 6174th
(six thousand one hundred and seventy-fourth)
Factorization 2 × 32× 73
Roman numeral VMCLXXIV
Binary 11000000111102
Ternary 221102003
Quaternary 12001324
Quinary 1441445
Senary 443306
Octal 140368
Duodecimal 36A612
Hexadecimal 181E16
Vigesimal F8E20
Base 36 4RI36

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

  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.

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 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 0 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:

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

9831 reaches 6174 after 7 iterations:

9831 – 1389 = 8442
8442 – 2448 = 5994
9954 – 4599 = 5355
5553 – 3555 = 1998
9981 – 1899 = 8082
8820 – 0288 = 8532 (rather than 882 – 288 = 594)
8532 – 2358 = 6174

4371 reaches 6174 after 7 iterations:

7431 - 1347 = 6084
8640 - 0468 = 8172 (rather than 864 - 468 = 396)
8721 - 1278 = 7443
7443 - 3447 = 3996
9963 - 3699 = 6264
6642 - 2466 = 4176
7641 - 1467 = 6174

8774, 8477, 8747, 7748, 7487, 7847, 7784, 4877, 4787, and 4778 reach 6174 after 4 iterations:

8774 – 4778 = 3996
9963 – 3699 = 6264
6642 – 2466 = 4176
7641 – 1467 = 6174

Note that in each iteration of Kaprekar's routine, the two numbers being subtracted one from the other have the same digit sum and hence the same remainder modulo 9. Therefore the result of each iteration of Kaprekar's routine is a multiple of 9.

Sequence of Kaprekar transformations ending in 6174

495 is the equivalent constant for three-digit numbers. For two-digit numbers, there is no equivalent constant; for any starting number with differing digits, the routine enters the loop (45, 9, 81, 63, 27, 45, … ). For each digit length greater than four, the routine may terminate at one of several fixed values or may enter one of several loops instead.[4]

Sequence of three digit Kaprekar transformations ending in 495

See also[edit]

References[edit]

  1. ^ Yutaka Nishiyama, 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. 
  4. ^ a b Weisstein, Eric W., "Kaprekar Routine", MathWorld.

External links[edit]