Kaprekar Number, Kaprekar made significant contributions by discovering various classes of numbers, such as Kaprekar numbers and the mysterious constant 6174, which possesses the unique property that the difference between its digits in Use the Kaprekar Number Calculator to check if a number is a Kaprekar number or generate Kaprekar numbers within a range. Example: 8531 Arrange the digits: In descending order (big to small): 8531 → 8531 In ascending order (small to big): 8531 → 1358 Subtract the smaller number from the bigger one: PDF | The (decimal) n-Kaprekar numbers are defined and are shown to be in one-one correspondence with the unitary divisors of 10^n - 1. Kaprekar introduced some interesting numbers; today, these numbers are known as Kaprekar Numbers (Kaprekar, 1980). For example: 45 2 = 2025, and 20+25 = 45, so 45 is a Kaprekar number. Tool to explore and visualize Kaprekar's algorithm, a fascinating mathematical routine involving sorting and subtracting numbers in a particular order to reveal constants like 6174 or 495. 95,121 = Kaprekar number: 95121 2 = 9048004641; 90480 + 04641 = 95121 [5] 95,420 = number of 22-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed [6] Kaprekar number Another class of numbers Kaprekar described are Kaprekar numbers. For example, in base 10, 45 is a 2-Kaprekar number, because 45² = 2025, and 20 + 25 = 45. The series of Kaprekar numbers is known as A006886, and begins as 1, 9, 45, 55, . The magic lies in a procedure that, when applied to almost any 4-digit number, will always lead to 6174. Thesis, Brunel Univ. Kaprekar is remembered today for a range of curious mathematical patterns that he discovered. . The best known is probably that related to the number 6174, sometimes called Kaprekar’s constant. Suppose that n2, when expressed in number base b, can be split into two parts that add up to n. Now compute and iterate. For example, 9 is a Kaprekar number since 9^2=81 8+1=9, and 297 is a Kaprekar number since 297^2=88209 88+209=297. , all four digit numbers excluding (0000, 1111, ) Sort four digits in ascending order and store result in a number "asc". 45 2 = 2025. International Journal of Number Theory (Pub. Find the square of the number n and separate the resulting number into two parts - right hand side with k digits and left hand side k or k-1 digits. The program will take a number as an input from the user and print if it is a Kaprekar number or not. May 20, 2025 · A Kaprekar number is a number whose square when divided into two parts and such that sum of parts is equal to the original number and none of the parts has value 0. 1. The number of iterations i needed for F p, b i (n) to reach a fixed point is the Kaprekar function's persistence of n, and undefined if it never reaches a fixed point. This video describes the simple math trick to obtain 495 from any four-digit number, using Kaprekar’s operation. This number is special as we always get this number when following steps are followed for any four digit number such that all digits of number are not same, i. If you square it, then split the number into two integers and sum those integers, you have the same value you started with. Another example is the number 703, its square Kaprekar number in Java explained with example, pros, cons, and real use cases. For each n >= 1 , we disallow 10 n as a Kaprekar number since it is the vacuous solution to (1) and (2) when N = 10 n . Kaprekar made remarkable discoveries that earned him a place in mathematical history. 인도의 어느 지역에 있는 철도의 선로 옆에는 $3025km$라고 쓰인 6174 is a very mysterious number. We square to arrive at a number that is either digits long or digits long. 45, since 45 2 =2025, and 20+25=45, also 9, 55, 99 etc. Lógicamente, si el número que se selecciona tiene todas sus cifras iguales, llevará a la constante trivial. Each number in the sequence uniquely determines the next number in the sequence. We will also write a program to find all Kaprekar numbers in between 1 to 10000. A Kaprekar number is a non-negative integer k such that when you square it and split the resulting number into two parts, the sum of those parts is equal to the original number k. Kaprekar. In particular, | Find, read and cite all the research you A number is said to be a kaprekar number if the square of the number is divided into two parts in such a way that the sum of divided parts equals the original number where none of the parts is having a value of 0. Kaprekar Constant (6174): Kaprekar is best known for the discovery of Kaprekar's con-stant, 6174. Java program to check if a number is Kaprekar number or not. In this piece you’ll learn how […] Kaprekar Routine A routine discovered in 1949 by D. In 1980, Indian mathematician D. In mathematics, a natural number in a given number base is a - Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add up to the original number. R. The process is iterative, such that the same procedure applies to the numbers obtained at each step. [10] A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e. Ömer Eğecioğlu and Bünyamin Şahin, On twin EP numbers, Transact. Add these two digits and if the result is as same as "n", then "n" is a Kaprekar Number. Note: A positive whole number ‘n’ that has ‘d’ number of digits is squared and split into two pieces… Kaprekar Sequences Wells' 1986 book [1] defines Kaprekar number roughly as follows: Definition A: N is a Kaprekar number if it has D digits, and if you take N2 and divide it into two pieces each D digits in size and add them together, you get N. See p. ) A Kaprekar number is a number whose square when divided into two parts and such that sum of parts is equal to the original number and none of the parts has value 0. Check if it is a Kaprekar number or not. Jan 23, 2025 · Kaprekar’s constant, 6174, is one of the most fascinating numbers in mathematics. 카프리카 수 (Karprekar number) 정수 중에는 특별한 이름이 붙은 수들이 많은데, 그 중에는 인도의 수학자 카프리카 (Kaprekar)의 이름이 붙은 카프리카 수 (Karprekar number)라는 것이 있는데, 카프리카 수는 다음과 같은 유래를 가지고 있다. Yutaka Nishiyama explains why, and how beautiful mathematical oddities can inspire us to discover new mathematics. Ever heard of a number that creates magic when squared? Meet the Kaprekar Number! 🧠 When you square it, then split the result into two parts, the sum adds Kaprekar numbers look like a party trick, yet they pack enough nuance to teach number theory, string slicing, data types, and test design in one sitting. The universal Kaprekar number 1 corresponds to the unitary divisor 1 of 10 n - 1, which is why we allow unity as a Kaprekar number. (UK, 2024). [4] In addition to the Kaprekar's constant and the Kaprekar numbers which were named after him, he also described self numbers or Devlali numbers, the harshad numbers and Demlo numbers. Sequence of Kaprekar Numbers The sequence of Kaprekar numbers begins: 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, … 2223 is a Kaprekar number, as 2223 ∗ 2223 = 4941729 , 4941729 may be split to 494 and 1729 , and 494 + 1729 = 2223 . Indian recreational mathematician D. The surprise is what this process leads to. Example 1: 45. Learn definition, logic, and program for better understanding. For example, in base 10, 45 is a 2-Kaprekar number, because 45 2 = 2025, and 20 + 25 = 45. Kaprekar in 1949, this number emerges from a simple process that always leads to the same result, no matter where you start. 20+25 = 45. Hence it is a Kaprekar Number. 카프리카 [1] 가 1955년 발견한 수로, 그의 이름을 따서 카프리카 수 라고 한다. Continuing with t Kaprekar number Let n be a k -digit integer in base b. Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers. カプレカー数 カプレカー数 （カプレカーすう、 英: Kaprekar number）とは、次のいずれかで定義される 自然数 である [1][2]。 2乗して上位と下位との数桁ずつに分けて和を取ったとき、元の値に等しくなる自然数。 Kaprekar's constant, or 6174, is a constant that arises when we take a 4-digit integer, form the largest and smallest numbers from its digits, and then subtract these two numbers. Since there are only finitely many possibilities, eventually the sequence must return to a number it hit before, leading to a cycle. Write a Program in Java to input a number and check whether it is a Kaprekar number or not. For 4-digit numbers (with at least two different digits), the routine involves: Kaprekar Number Indian mathematician Dattatreya Ramchandra Kaprekar discovered this amazing trick. Santanu Bandyopadhyay, Kaprekar Number, Indian Institute of Technology Bombay (Mumbai, India, 2020). Now given a n A modified Kaprekar number is a positive whole number with a special property. カプレカー数 カプレカー数 （カプレカーすう、 英: Kaprekar number）とは、次のいずれかで定義される 自然数 である [1][2]。 2乗して上位と下位との数桁ずつに分けて和を取ったとき、元の値に等しくなる自然数。 International Journal of Number Theory (Pub. To apply the Kaprekar routine to a number , arrange the digits in descending () and ascending () order. If we construct the largest 4-digit number using only the digits Kaprekar number In mathematics, a natural number in a given number base is a - Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add up to the original number. Explore examples and step-by-step calculations. e. 4879 2 = 23804641. Discovered by Indian mathematician D. Steps: Select any 4-digit number (where no digits are the same). 142. Given a number N. In mathematics, a Kaprekar number (By: Shri Dattatreya Ramachandra Kaprekar, 1905–1986, Indian mathematician) is that non-negative integer such that, in a given base, the digits of its square on that basis they can be separated into two numbers which added together give the original number. 인도의 어느 지역에 있는 철도의 선로 옆에는 $3025km$라고 쓰인 i < j n, what is ja1 b1j + ja2 b2j + :::: + jan bnj? R Kaprekar (1905 to 1986) was an Indian mathematician known for his signi cant con-tributions to recreational mathematics. A Kaprekar number is a sociable Kaprekar number with k = 1, and a amicable Kaprekar number is a sociable Kaprekar number with k = 2. 238+04641 = 4879. Kaprekar process for three digit numbers converging to 495 [Wikimedia Commons]. g. Note:- A Kaprekar number is a number whose square when divided into two parts the sum of those parts is equal to the original number and none of the parts has value 0. by World Scientific) p-Adic Numbers, Ultrametric Analysis and Applications (Pleiades Pub, Russian Academy of Sciences, distributed by Springer) Kaprekar’s Constant: The Mathematical Mystery of 6174 Mathematics is full of interesting patterns, but few are as intriguing as 6174, also known as Kaprekar’s Constant. C programming exercises and solution: Write a program in C to check whether a given number is a Kaprekar number or not. by World Scientific) p-Adic Numbers, Ultrametric Analysis and Applications (Pleiades Pub, Russian Academy of Sciences, distributed by Springer) The Kaprekar constant is a special number in number theory related to a process called Kaprekar's routine. Kaprekar for 4-digit numbers, but which can be generalized to -digit numbers. The simplest example is 9, its square is 81 and 8+1= 9. For 4-digit numbers (with at least two different digits), the routine involves: Dattatreya Ramachandra Kaprekar discovered 6174 as the magical constant for 4-digit numbers. I still remember the first time I saw 45² = 2025 and noticed that 20 + 25 equals 45—my mental debugger lit up. Kaprekar Number Indian mathematician Dattatreya Ramchandra Kaprekar discovered this amazing trick. 5 days ago · If the resultant sum is k, then k is called a Kaprekar number. This number arises from an iterative Kaprekar number In mathematics, a non-negative integer is called a "Kaprekar number" for a given base if the representation of its square in that base can be split into two parts that add up to the original number, with the proviso that the part formed from the low-order digits of the square must be non-zero—although it is allowed to include Kaprekar Number : Consider "n" as a positive integer that has k number of digits. The complete mathematical explanation with examples is there. The Magic Number of Kaprekar "The Magic Number of Kaprekar," also known as Kaprekar's constant, is the number 6174. Kaprekar published a surprising result, concerning what is now known as Kaprekar's process. It is part of a fascinating mathematical process discovered in 1949 by a school teacher from Maharashtra named D. Example 2: 4879. Mar 16, 2025 · Mathematics is full of interesting patterns, but few are as intriguing as 6174, also known as Kaprekar’s Constant. What makes the Kaprekar numbers curious and interesting? Let's consider an example. 6174 is a very mysterious number. D. Kaprekar in the 1940s, this four-digit constant has captivated math enthusiasts and casual learners alike. 개요 [편집] Kaprekar number (constant) · Kaprekar 數 (常 數) 인도 의 수학자 D. Here is an example of a Kaprekar number for cubes: 7172 3 = 368910352448 and Kaprekar's Iterations and Constants In 1955, D. Como aplicar la operación de Kaprekar a 6174 da 6174, ese número es un punto fijo; y por tanto, una constante de Kaprekar El cero es una constante de Kaprekar trivial para todos los casos, debido a que . Jun 27, 2025 · What Makes a Number “Kaprekar”? A Kaprekar number has a unique property: when you square it and then split the result into two parts, the sum of those parts equals the original number. 297 is a Kaprekar number: square of 297 is 88209; divided it into two portions, 88 and 209; now total them, Kaprekar Number is a number whose square when divided into two parts such that their sum equals to the original number. The Kaprekar constant is a special number in number theory related to a process called Kaprekar's routine. Example: 8531 Arrange the digits: In descending order (big to small): 8531 → 8531 In ascending order (small to big): 8531 → 1358 Subtract the smaller number from the bigger one: 6174 is the Kaprekar Constant. Then Arrange the digits of the number in the increasing and also in the A Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again. We will learn what is a Kaprekar number, how to check if a number is a Kaprekar number or not using C. So, start with any 4-digit number. Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. Then n is a Kaprekar number for base b. Then n is said to be a Kaprekar number in base b if n 2 has the following property: when you add the number formed by its right hand digits to that formed by its left hand digits, you get n. Consider a positive whole number with digits. R. Discovered by the Indian mathematician D. In mathematics, a natural number in a given number base is a - Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has digits, that add up to the original number. o9pk, ahhscx, sifl, fyavf, o4p6oo, i6wr6, ijps, uwmr, fhtj, psnu,