6174 (number)
Encyclopedia
6174 is known as Kaprekar's constant after the India
n mathematician
D. R. Kaprekar
. This number is notable for the following property:
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 3524:
The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigit
s 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:
9831 reaches 6174 after 7 iterations:
8774, 8477, 8747, 7748, 7487, 7847, 7784, 4877, 4787, and 4778 reach 6174 after 4 iterations:
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.
495
is the equivalent constant for three-digit numbers. For five-digit numbers and above, there is no single equivalent constant; for each digit length the routine may terminate at one of several fixed values or may enter one of several loops instead.
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
D. R. Kaprekar
D. R. Kaprekar
Dattaraya Ramchandra Kaprekar was an Indian mathematician who discovered several results in number theory, including a class of numbers and a constant named after him...
. This number is notable for the following property:
- Take any four-digit number, using at least two different digits. (Leading zeros are allowed.)
- Arrange the digits in ascending and then in descending 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.
The above process, known as Kaprekar's routine, will always reach its fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
, 6174, in at most 7 iterations. 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
The only four-digit numbers for which Kaprekar's routine does not reach 6174 are repdigit
Repdigit
In recreational mathematics, a repdigit is a natural number composed of repeated instances of the same digit, most often in the decimal numeral system....
s such as 1111, which give the result 0
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
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
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.
495
495 (number)
495 is the integer after 494 and before 496. It is a pentatope number.- Kaprekar transformation :The Kaprekar transformation is defined as follows for three-digit numbers:# Start with a three-digit number with at least two digits different....
is the equivalent constant for three-digit numbers. For five-digit numbers and above, there is no single equivalent constant; for each digit length the routine may terminate at one of several fixed values or may enter one of several loops instead.