# 6174 (number)

(Redirected from Kaprekar's constant)
 ← 6173 6174 6175 →
Cardinalsix thousand one hundred seventy-four
Ordinal6174th
(six thousand one hundred seventy-fourth)
Factorization2 × 32 × 73
Greek numeral,ϚΡΟΔ´
Roman numeralVMCLXXIV
Binary11000000111102
Ternary221102003
Quaternary12001324
Quinary1441445
Senary443306
Octal140368
Duodecimal36A612
VigesimalF8E20
Base 364RI36

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

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 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 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"

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".

## Other properties

• 6174 is a Harshad number, since it is divisible by the sum of its digits:
• 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 degrees of 18:
183 + 182 + 181 = 5832 + 324 + 18 = 6174.
• 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.[importance?]

## References

1. ^
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. ^