6174 – Wikipedia
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The quantity 6174 is named Kaprekar’s fixed[1][2][3] after the Indian mathematician D. R. Kaprekar. This quantity is famend for the next rule:
- Take any four-digit quantity, utilizing a minimum of two totally different digits (main zeros are allowed).
- Organize the digits in descending after which in ascending order to get two four-digit numbers, including main zeros if mandatory.
- Subtract the smaller quantity from the larger quantity.
- Return to step 2 and repeat.
The above course of, often called Kaprekar’s routine, will at all times attain its fixed point, 6174, in at most 7 iterations.[4] As soon as 6174 is reached, the method will proceed yielding 7641 – 1467 = 6174. For instance, select 1459:
- 9541 – 1459 = 8082
- 8820 – 0288 = 8532
- 8532 – 2358 = 6174
- 7641 – 1467 = 6174
The one four-digit numbers for which Kaprekar’s routine doesn’t attain 6174 are repdigits equivalent to 1111, which give the outcome 0000 after a single iteration. All different four-digit numbers finally attain 6174 if main zeros are used to maintain the variety of digits at 4. For numbers with three an identical numbers and a fourth quantity that’s one quantity increased or decrease (equivalent to 2111), it’s important to deal with 3-digit numbers with a number one zero; for instance: 2111 – 1112 = 0999; 9990 – 999 = 8991; 9981 – 1899 = 8082; 8820 – 288 = 8532; 8532 – 2358 = 6174.[5]
Pure quantity
| Cardinal | six thousand 100 seventy-four |
|---|---|
| Ordinal | 6174th (six thousand 100 seventy-fourth) |
| Factorization | 2 × 32 × 73 |
| Divisors | 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 343, 441, 686, 882, 1029, 2058, 3087, 6174 |
| Greek numeral | ,ϚΡΟΔ´ |
| Roman numeral | VMCLXXIV, or VICLXXIV |
| Binary | 11000000111102 |
| Ternary | 221102003 |
| Senary | 443306 |
| Octal | 140368 |
| Duodecimal | 36A612 |
| Hexadecimal | 181E16 |
Different “Kaprekar’s constants”[edit]
There could be analogous mounted factors for digit lengths aside from 4; as an example, if we use 3-digit numbers, then most sequences (i.e., aside from repdigits equivalent to 111) will terminate within the worth 495 in at most 6 iterations. Typically these numbers (495, 6174, and their counterparts in different digit lengths or in bases aside from 10) are known as “Kaprekar constants”.
Different properties[edit]
- 6174 is a 7-smooth number, i.e. none of its prime elements are higher than 7.
- 6174 could be written because the sum of the primary three powers of 18:
- 183 + 182 + 181 = 5832 + 324 + 18 = 6174, and coincidentally, 6 + 1 + 7 + 4 = 18.
- The sum of squares of the prime elements of 6174 is a sq.:
- 22 + 32 + 32 + 72 + 72 + 72 = 4 + 9 + 9 + 49 + 49 + 49 = 169 = 132
References[edit]
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