Skip to main content

Kaprekar Number

Kaprekar number is one of the amazing concepts in Mathematics.

The square of a number when divided into parts and none of the parts is equal to 0, and sum of both the parts forms the original number, then that number is a Kaprekar number.

Let's understand this concept in detail:

Checking a Kaprekar Number

Following are the steps to check whether any number is a Kaprekar number or not:

Step 1: Consider any number N (must be greater than 0).

Step 2: Square the number N (N x N).

Step 3: Divide the resultant obtained in Step 2 into two parts (left and right).

Possible scenarios that occur when dividing the resultant into two parts:

Scenario 1: If the resultant obtained in Step 2 is a single digit number

If the resultant obtained in Step 2 is a single digit number (no matter even or odd), compare it with the number considered in Step 1. If the number in Step 2 is equal to the number in Step 1, it is a Kaprekar number. But if the number in Step 2 is not equal to the number in Step 1, it is not a Kaprekar number.

Scenario 2: If the total number of digits obtained in Step 2 is even

If the total number of digits obtained in Step 2 is even, divide into two equal parts (left and right).

For example, the number 1234 have total 4 digits (even). Divide it into two equal parts as 12 (left part) and 34 (right part).

Scenario 3: If the total number of digits obtained in Step 2 is odd

If the total number of digits obtained in Step 2 is odd, divide into two parts (left and right). The left part contains n-1 number of digits, and the right part contains n number of digits.

For example, the number 12345 have total 5 digits (odd). Divide it into two parts as 12 (left part) and 345 (right part).

Step 4: Sum both parts (left + right).

Step 5: If the resultant obtained in Step 4 is equal to the number considered in Step 1, the number is a Kaprekar number. But if the resultant is not equal to the number considered in Step 1, the number is not a Kaprekar number.

Examples

First Example

Step 1: Consider number 1.

Step 2: Square the number 1 (1 x 1), the resultant is 1.

Step 3: Divide the resultant obtained in Step 2 into two parts (left and right).

The resultant obtained in Step 2 is a single digit number and also equal to the number considered in Step 1. Hence, number 1 is a Kaprekar number.

Second Example

Step 1: Consider number 5.

Step 2: Square the number 5 (5 x 5), the resultant is 25.

Step 3: Divide the resultant obtained in Step 2 into two parts (left and right).

The number of digits obtained in Step 2 is even. Dividing into two equal parts gives the resultant as 2 (left part) and 5 (right part).

Step 4: Sum both parts (left + right), that is, 2 + 5 = 7.

Step 5: The resultant obtained in Step 4 is not equal to the number considered in Step 1, that is, 7 is not equal to 5. Hence, number 5 is not a Kaprekar number.

Third Example

Step 1: Consider number 9.

Step 2: Square the number 9 (9 x 9), the resultant is 81.

Step 3: Divide the resultant obtained in Step 2 into two parts (left and right).

The number of digits obtained in Step 2 is even. Dividing into two equal parts gives the resultant as 8 (left part) and 1 (right part).

Step 4: Sum both parts (left + right), that is, 8 + 1 = 9.

Step 5: The resultant obtained in Step 4 is equal to the number considered in Step 1, that is, 9 is equal to 9. Hence, number 9 is a Kaprekar number.

Hope you have understood this amazing concept of the Kaprekar number. You may try yourself using any number and check whether it is a Kaprekar number or not. In case of any query, please feel free to provide your valuable comments.

Labels:

Comments

Post a Comment

Popular posts from this blog

The Number 1.6: Base of Fibonacci Sequence and the Golden Ratio

This post is about the number 1.6 that forms the base of the mathematical concepts Fibonacci sequence and the Golden ratio. For better understanding of this amazing number 1.6, it is important to know about Fibonacci sequence and the Golden ratio. Golden Ratio The  Golden ratio is the ratio of a line segment cut into two pieces of distinct lengths such that the ratio of the longer segment to the shorter segment is equal to the whole segment to that of the longer segment. It is denoted by the symbol Phi [ø]. Suppose there is a line segment AB, and C is any point that divides the line segment AB into two pieces of distinct lengths such that AC is greater than CB. Then, as per the definition, [ø] = AC/CB = AB/AC. The value of the Golden ratio [ø] is 1.6 1803398875 . As you can see, the value of the Golden ratio starts with the number 1.6. For more information on the Golden ratio, see https://theamazingconcepts.blogspot.com/2021/07/the-golden-ratio-key-to-amazing-design....
Post a Comment
Read more

Kaprekar Constant 495 for Three-digit Number

Dattatreya Ramchandra Kaprekar (D. R. Kaprekar; 1905 – 1986) was an Indian mathematician who discovered several classes of natural numbers including Kaprekar Constant. The number  495  is called Kaprekar Constant for three-digit number. Why is the number 495 called Kaprekar Constant? In 1949, D. R. Kaprekar discovered an interesting property of the number 495 through the mathematical calculations. D. R. Kaprekar observed that when any three-digit number of non-identical digits (not all digits are same) is arranged in two parts – highest number and lowest number, and difference is calculated using subtraction method, then the end-result will be always 495. If the process is continued even after getting the end-result 495, then the end-result will be always 495 at each Iteration, therefore the number 495 is known as Kaprekar Constant. Steps to Follow Take any three-digit number with different digits. Form the highest number and lowest number from this three-digit number. Fi...
Post a Comment
Read more

Harshad Number or Niven Number

The concept of Harshad number was given by D. R. Kaprekar, one of the great mathematicians from the country India. Harshad number or Niven number is any positive integer b that is divisible by its sum of digits (SOD). The term Niven number was coined by Ivan Morton Niven, a Canadian-American mathematician, who specializes in number theory. As per the definition: Harshad number b = number / sum of digits Below are some examples: Number 12 Sum of digits = 1 + 2 => 3 Here, number 12 is divisible by its sum of digits 3, and the resultant is 4. Number 18 Sum of digits = 1 + 8 => 9 Here, number 18 is divisible by its sum of digits 9, and the resultant is 2. Number 72 Sum of digits = 7 + 2 => 9 Here, number 72 is divisible by its sum of digits 9, and the resultant is 8. Number 110 Sum of digits = 1 + 1 + 0 => 2 Here, number 110 is divisible by its sum of digits 2, and the resultant is 55. Number 135 Sum of digits = 1 + 3 + 5 => 9 Here, number...
Post a Comment
Read more