Use numerical evidence to make conjectures about the behavior of the sequence a, T(a), T(T(a)), … where a is a five-digit integer in base 10 notation in which not all digits are the same, and T(a) is defined as in the preamble to Exercise.
Exercise
Let a be an integer with a four-digit decimal expansion, where not all digits are the same. Let a′ be the integer with a decimal expansion obtained by writing the digit of a in descending order, and let a″ be the integer with a decimal expansion obtained by writing the digits of a in ascending order. Define T(a) = a′ − a″. For instance, T(7318) = 8731 − 1378 = 7353.
Show that the only integer with a four-digit decimal expansion (where not all digits are the same) such that T(a) = a is a = 6174. The integer 6174 is called Kaprekar’s constant, after the Indian mathematician D.R.Kaprekar, because it is the only integer with this property.
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