6174 (number)

← 6173 6174 6175 →
← 0 [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]] [[{{#expr:{{{1}}}*{{{factor}}}*1000}} (number)|{{#ifeq:{{{1}}}|10|→|{{#expr:{{{1}}}*{{{factor}}}}}k}}]]
Cardinal six thousand one hundred seventy-four
Ordinal 6174th
(six thousand one hundred 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 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"

Note that there can be analogous fixed points for digit lengths other than four, for instance if we use 3-digit numbers then all sequences 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 divided by the sum of its digits:

6174 = (1 + 6 + 7 + 4) × 343.

6174 is a 7-smooth number, i. e. all its prime factors are not greater than 7.

6174 is a practical number , because an arbitrary number less than 6174 can be represented as a sum of various factors of the number 6174. This is a not uncommon property, and the nearest neighboring practical numbers are 6160, 6162, 6180, 6188.

6174 can be written as the sum of the first three degrees of the natural number 18:

18³ + 18² + 18 = 5832 + 324 + 18 = 6174.

The sum of squares of the prime factors of 6174 is an exact square:

2² + 3² + 3² + 7² + 7² + 7² = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 13².

References

  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. ↑ Weisstein, Eric W. "Kaprekar Routine". MathWorld.
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