Kaprekar Number Calculator
Check if a number is a Kaprekar number by squaring it and finding a split where the two parts sum to the original number.
Enter any positive integer (up to 15 digits) to check if it is a Kaprekar number
A Kaprekar number is a positive integer K such that when K² is split into two parts, those parts sum back to K. The right part cannot be all zeros.
Algorithm:
- Square the input number K to get K²
- Try all possible splits of K² into left and right parts
- Check if left + right = K for any valid split
- The right part cannot consist entirely of zeros
Examples:
- 45² = 2025 → 20 + 25 = 45 (Kaprekar)
- 297² = 88209 → 88 + 209 = 297 (Kaprekar)
- 703² = 494209 → 494 + 209 = 703 (Kaprekar)
- 10² = 100 → no valid split sums to 10 (Not Kaprekar)
Mathematical Property: All numbers of the form 10ⁿ - 1 (i.e., 9, 99, 999, 9999, ...) are Kaprekar numbers. This is because (10ⁿ - 1)² = 10²ⁿ - 2×10ⁿ + 1, which always splits to give a sum equal to the original number.
History: These numbers are named after Indian mathematician D.R. Kaprekar (1905-1986), who discovered this property while working as a schoolteacher in Devlali, India. He made several contributions to recreational mathematics, including the famous Kaprekar constant 6174.
Applications: Kaprekar numbers are primarily of interest in recreational mathematics and number theory. They demonstrate interesting properties of the decimal number system and provide engaging puzzles for mathematical exploration.
What is a Kaprekar number?
A Kaprekar number is a positive integer K with the property that when K is squared, the resulting number can be split into two parts that sum back to K. For example, 45 is a Kaprekar number because 45² = 2025, and 20 + 25 = 45. The concept is named after Indian mathematician D.R. Kaprekar who discovered it in 1949.
What are some examples of Kaprekar numbers?
The first several Kaprekar numbers are: 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999. For instance, 9² = 81 and 8 + 1 = 9; 297² = 88209 and 88 + 209 = 297; 703² = 494209 and 494 + 209 = 703.
How do you check if a number is a Kaprekar number?
To check if a number K is a Kaprekar number: (1) Square the number to get K². (2) Try all possible ways to split K² into a left part and a right part. (3) If any split results in two parts that sum to K, then K is a Kaprekar number. The right part cannot consist entirely of zeros.
Who discovered Kaprekar numbers?
Kaprekar numbers are named after Dattatreya Ramchandra Kaprekar (1905-1986), an Indian recreational mathematician who made several contributions to number theory. He worked as a schoolteacher in Devlali, India, and discovered many interesting number properties including the Kaprekar constant (6174) and Kaprekar numbers.
Are all numbers of the form 10^n - 1 Kaprekar numbers?
Yes! All numbers consisting entirely of 9s (like 9, 99, 999, 9999, etc.) are Kaprekar numbers. For example, 99² = 9801, and 98 + 01 = 99. This pattern holds because (10^n - 1)² = 10^(2n) - 2×10^n + 1, which splits perfectly.