Input a number to check if it's a Kaprekar number, or enter a range to generate all Kaprekar numbers within it.
Kaprekar Number Check or Generate
Result
Result
What is a Kaprekar Number?
A Kaprekar number is a special type of number. For a positive integer \( n \) to qualify as a Kaprekar number:
Square the number: \( n^2 \).
Split the resulting square into two parts (left and right).
Add the two parts together.
If the sum equals the original number \( n \), then \( n \) is a Kaprekar number. For example: \( 9^2 = 81 \). Split \( 81 \) into \( 8 \) and \( 1 \). \( 8 + 1 = 9 \), so \( 9 \) is a Kaprekar number.
How to Determine a Kaprekar Number?
Calculate the square: Compute the square of the number \( n^2 \).
Split the square: Divide the digits of the square into two parts. The left part represents the higher digits, and the right part represents the lower digits.
Sum the parts: Add the left and right parts together.
Compare the result: If the sum equals the original number \( n \), it's a Kaprekar number. Otherwise, it's not.
Examples
Example 1: Is 45 a Kaprekar Number?
Solution:
1. Calculate the square:
\(45^2 = 2025\)
2. Split the square:
Split \( 2025 \) into \( 20 \) and \( 25 \).
3. Sum the parts:
\(20 + 25 = 45\)
Result:
\( 45 \) is a Kaprekar number because the sum matches the original number.
Example 2: Is 13 a Kaprekar Number?
Solution:
1. Calculate the square:
\(13^2 = 169\)
2. Split the square:
Possible splits are \( 1 \) and \( 69 \), or \( 16 \) and \( 9 \).
3. Sum the parts:
\( 1 + 69 = 70 \) and \( 16 + 9 = 25 \).
Result:
Neither sum equals \( 13 \), so \( 13 \) is not a Kaprekar number.
Example 3: Is 99 a Kaprekar Number?
Solution:
1. Calculate the square:
\(99^2 = 9801\)
2. Split the square:
Split \( 9801 \) into \( 98 \) and \( 01 \).
3. Sum the parts:
\(98 + 01 = 99\)
Result:
\( 99 \) is a Kaprekar number because the sum matches the original number.