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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.

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