**Kaprekar number**

Named
after Dattaraya Ramchandra Kaprekar .

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.

**Examples:**

297
is a Kaprekar number for base 10, because 297² = 88209, which can be split into
88 and 209, and 88 + 209 = 297.

45
is a Kaprekar number, because 45² = 2025 and 20+25 = 45.

Let X be a non-negative integer.

X is a Kaprekar number for base b if
there exist non-negative integers n, A, and positive number B satisfying:

X² = Abn + B, where 0 < B < bn

X = A + B

Note that X is also a Kaprekar
number for base bn, for this specific choice of n. More narrowly, we can define
the set K(N) for a given integer N as the set of integers X for which[1]

X² = AN + B, where 0 < B < N

X = A + B

Each Kaprekar number X for base b is
then counted in one of the sets K(b), K(b²), K(b³),….

**Note:**

By convention, the second part may
start with the digit 0, but must be nonzero.

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 is zero.

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