Kaprekar number facts for kids

Imagine a special kind of natural number in mathematics! It's called a Kaprekar number. These numbers have a cool trick: when you square them, you can split the result into two parts. If you add these two parts together, you get the original number back!

For example, let's look at the number 45.

• First, we square 45: 45 x 45 = 2025.
• Next, we split 2025 into two parts: 20 and 25.
• Finally, we add these parts: 20 + 25 = 45.

Since we got 45 back, 45 is a Kaprekar number! These numbers are named after an Indian mathematician, D. R. Kaprekar.

What are Kaprekar Numbers?

A Kaprekar number is a positive whole number that has this unique property. It works in different "number bases," which are like different ways of counting. Most of the time, we use base 10 (our regular counting system with digits 0-9).

How Do They Work?

To find out if a number is a Kaprekar number, you follow these steps:

• Step 1: Square the number. Multiply the number by itself.
• Step 2: Split the squared number. You divide the squared number into two pieces. The second piece must have the same number of digits as the original number. For example, if your original number was 45 (two digits), the second part of its square (2025) must also have two digits (25).
• Step 3: Add the two parts. If the sum of these two parts equals your original number, then it's a Kaprekar number!

Let's try another example in base 10. Is 9 a Kaprekar number?

• Square 9: 9 x 9 = 81.
• Split 81: Since 9 has one digit, the second part of 81 needs one digit. We can split 81 into 8 and 1.
• Add the parts: 8 + 1 = 9.

Yes, 9 is also a Kaprekar number!

Simple Examples

The numbers 0 and 1 are considered "trivial" Kaprekar numbers. This means they are very simple cases.

• 0 squared is 0. Split 0 into 0 and 0. 0 + 0 = 0.
• 1 squared is 1. Split 1 into 0 and 1. 0 + 1 = 1.

All other Kaprekar numbers are called "nontrivial."

Kaprekar Numbers in Different Number Systems

Numbers can be written in different "bases." Our everyday system is base 10. But there are other bases, like base 2 (used by computers) or base 16. Kaprekar numbers exist in these other bases too!

For example, in base 2, some special numbers are Kaprekar numbers.

Exploring Kaprekar Numbers and Cycles

Sometimes, if a number isn't a Kaprekar number itself, it might lead to one after a few steps. Or, it might get stuck in a "cycle" where it keeps repeating a sequence of numbers.

Here are some examples of Kaprekar numbers in different bases and with different "powers" (which relates to how many digits you split the number into). Remember, these numbers are shown in their specific base, not always base 10!

Base	Power	Nontrivial Kaprekar numbers ,	Cycles
2	1	10
3	1	2, 10
4	1	3, 10
5	1	4, 10
6	1	5, 10
7	1	3, 4, 6, 10
8	1	7, 10	2 → 4 → 2
9	1	8, 10
10	1	9, 10
11	1	5, 6, A, 10
12	1	B, 10
13	1	4, 9, C, 10
14	1	D, 10
15	1	7, 8, E, 10	2 → 4 → 2 9 → B → 9
16	1	6, A, F, 10
2	2	11
3	2	22, 100
4	2	12, 22, 33, 100
5	2	14, 31, 44, 100
6	2	23, 33, 55, 100	15 → 24 → 15 41 → 50 → 41
7	2	22, 45, 66, 100
8	2	34, 44, 77, 100	4 → 20 → 4 11 → 22 → 11 45 → 56 → 45
10	2	45, 55, 99, 100
2	3	111, 1000	10 → 100 → 10
3	3	111, 112, 222, 1000	10 → 100 → 10
2	4	110, 1010, 1111, 10000
3	4	121, 2102, 2222, 10000
2	5	11111, 100000	10 → 100 → 10000 → 1000 → 10 111 → 10010 → 1110 → 1010 → 111
3	5	11111, 22222, 100000	10 → 100 → 10000 → 1000 → 10
2	6	11100, 100100, 111111, 1000000	100 → 10000 → 100 1001 → 10010 → 1001 100101 → 101110 → 100101
3	6	10220, 20021, 101010, 121220, 202202, 212010, 222222, 1000000	100 → 10000 → 100 122012 → 201212 → 122012
2	7	1111111, 10000000	10 → 100 → 10000 → 10 1000 → 1000000 → 100000 → 1000 100110 → 101111 → 110010 → 1010111 → 1001100 → 111101 → 100110
3	7	1111111, 1111112, 2222222, 10000000	10 → 100 → 10000 → 10 1000 → 1000000 → 100000 → 1000 1111121 → 1111211 → 1121111 → 1111121
2	8	1010101, 1111000, 10001000, 10101011, 11001101, 11111111, 100000000
3	8	2012021, 10121020, 12101210, 21121001, 20210202, 22222222, 100000000
2	9	10010011, 101101101, 111111111, 1000000000	10 → 100 → 10000 → 100000000 → 10000000 → 100000 → 10 1000 → 1000000 → 1000 10011010 → 11010010 → 10011010

Beyond Positive Numbers

Mathematicians have even explored how Kaprekar numbers might work with negative integers. This involves using special ways to represent numbers that include both positive and negative digits.

See also

• Arithmetic dynamics
• Automorphic number
• Dudeney number
• Factorion
• Happy number
• Kaprekar's constant
• Meertens number
• Narcissistic number
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Sum-product number