Kaprekar number Totally Explained

Everything about Kaprekar Number totally explained

In mathematics, 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. For example, 297 is a Kaprekar number for base 10, because 297² = 88209, which can be split into 88 and 209, and 88 + 209 = 297. The second part may start with the digit 0, but must be positive. For example, 999 is a Kaprekar number for base 10, because 999² = 998001, which can be split into 998 and 001, and 998 + 001 = 999. But 100 is not; although 100² = 10000 and 100 + 00 = 100, the second part here isn't positive. Stated mathematically, let X be a non-negative integer. X is a Kaprekar number for base b if there exist non-negative integers n, A and B satisfying the following three conditions:
ť 0 < B < bⁿ

    X² = Abⁿ + B ť X = A + B

The first few Kaprekar numbers in base 10 are :
ť 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170

In binary, all even perfect numbers are Kaprekar numbers.
   For any base there exist infinitely many Kaprekar numbers; in particular, for base b all numbers of the form bⁿ - 1 are Kaprekar numbers.
   The Kaprekar numbers are named after D. R. Kaprekar.

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