Kaprekar's contants are the numbers (or cycle of numbers) that occur when the digits of a number are arranged in descending and ascending order, subtracted from one another, and the procedure repeated until a constant or cycle of constants occurs. For example, take the number 1835. 8531 - 1358 = 7173 7731 - 1377 = 6354 6543 - 3456 = 2A87 A872 - 278A = 7A82 So 7A82 is one Kaprekar constant for a 4 digit number using base ten. As you may recall from my articles, Kaprekar's constant in ANS, with 4 digits in base A (ten), now includes the number 7A82 as well as 6174. It's interesting to examine the factors of these numbers. 6174 = 2 x 3 x 3 x 7 x 7 x 7 7A82 = 2 x 3 x 3 x 449 Notice that both numbers have 18 = 2 x 3 x 3 as a common factor. The digit sums of these in base ten are 18 and 27 respectively, and 27 = 3 x 3 x 3. In base ten, there is only one three digit constant : 495 = 3 x 3 x 5 x 11. More things to explore : (1) In base ten using a different number of digits, what happens? (2) In ENS, cycles of constants also occur. What happens in ANS? (3) What constants/cycles occur in other bases? (4) Why do these constants and cycles occur? (5) What determines the factors of these constants? (6) Why does only one number occur for some, multiples for others and cycles for others?
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