Verify the result described in Exercise for several different four-digit integers, in which not all digits are the same.
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.
a) Show that if a is a positive integer with a four-digit decimal expansion where not all digits are the same, then the sequence a, T(a), T(T(a)),T(T(T(a))),… , obtained by iterating T, eventually reaches the integer 6174.
b) Determine the maximum number of steps required for the sequence defined in part (a) to reach 6174.
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