495 (number)

495 (four hundred [and] ninety-five) is the natural number following 494 and preceding 496. It is a pentatope number^[1] (and so a binomial coefficient ${\displaystyle {\tbinom {12}{4}}}$). The maximal number of pieces that can be obtained by cutting an annulus with 30 cuts.^[2]

← 494 495 496 →
List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	four hundred ninety-five
Ordinal	495th (four hundred ninety-fifth)
Factorization	3^2 × 5 × 11
Greek numeral	ΥϞΕ´
Roman numeral	CDXCV
Binary	1111011112
Ternary	2001003
Senary	21436
Octal	7578
Duodecimal	35312
Hexadecimal	1EF16

Kaprekar transformation

The Kaprekar's routine algorithm is defined as follows for three-digit numbers:

1. Take any three-digit number, other than repdigits such as 111. Leading zeros are allowed.
2. Arrange the digits in descending and then in ascending order to get two three-digit numbers, adding leading zeros if necessary.
3. Subtract the smaller number from the bigger number.
4. Go back to step 2 and repeat.

Repeating this process will always reach 495 in a few steps. Once 495 is reached, the process stops because 954 – 459 = 495.

Example

For example, choose 495:

495

The only three-digit numbers for which this function does not work are repdigits such as 111, which give the answer 0 after a single iteration. All other three-digit numbers work if leading zeros are used to keep the number of digits at 3:

211 – 112 = 099

990 – 099 = 891 (rather than 99 – 99 = 0)

981 – 189 = 792

972 – 279 = 693

963 – 369 = 594

954 − 459 = 495

The number 6174 has the same property for the four-digit numbers, albeit has a much greater percentage of workable numbers. Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Peyush constants" named after Peyush Dixit who solved this routine as a part of his IMO 2000 (International Mathematical Olympiad, Year 2000) thesis. ^[3]

See also

• Collatz conjecture — sequence of unarranged-digit numbers always ends with the number 1.

References

1. "Sloane's A000332". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-16.
2. Sloane, N. J. A. (ed.). "Sequence A000096". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
3. Hanover 2017, p. 14, Operations.

• Eldridge, Klaus E.; Sagong, Seok (February 1988). "The Determination of Kaprekar Convergence and Loop Convergence of All Three-Digit Numbers". The American Mathematical Monthly. 95 (2). The American Mathematical Monthly, Vol. 95, No. 2: 105–112. doi:10.2307/2323062. JSTOR 2323062.