Numbers: Their Tales, Types, and Treasures.

Chapter 8: Special Numbers

 

8.6.KAPREKAR NUMBERS

There are other numbers that have unusual peculiarities as well. Sometimes these peculiarities can be understood and justified through an algebraic representation, while at other times a peculiarity is simply a quirk of the base-10 number system. In any case, these numbers provide us with some rather entertaining amusements that ought to motivate us to look for other such peculiarities or oddities.

Consider, for example, the number 297. When we take the square of that number, we get 2972 = 88,209, and, strangely enough, if we were to split it up into two numbers, the sum of the two numbers results in the original number: 88 + 209 = 297. Such a number is called a Kaprekar number, named after the Indian mathematician Dattaraya Ramchandra Kaprekar (1905–1986) who discovered such numbers. Here are a few more examples:

92 = 81…8 + 1 = 9
452 =2025…20 + 25 = 45
552 = 3025…30 + 25 = 55
7032 = 494,209…494 + 209 =703
2,7282 = 7,441,984…744 + 1,984 = 2,728
4,8792 = 23,804,641…238 + 04,641 = 4,879
142,8572 = 20,408,122,449…20,408 + 122,449 = 142,857

A more comprehensive table is given in the appendixsection 5. Some higher Kaprekar numbers are: 38,962; 77,778; 82,656; 95,121; 99,999;…538,461; 857,143….

There are also further variations, such as the number 45, which we would consider a Kaprekar triple, since it behaves as follows: 453 = 91,125 = 9 + 11 + 25 = 45. Other Kaprekar triples are: 1, 8, 10, 297, and 2322. Curiously enough, the number 297, which we previously demonstrated as a Kaprekar number, is also a Kaprekar triple, since 2973 = 26,198,073, and 26 + 198 + 073 = 297. Readers may choose to find other Kaprekar triples.