Kaprekar's Process on 5 Digit Number

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Theorem

Let $n$ be a $5$-digit integer whose digits are not all the same.

Kaprekar's process, when applied to $n$, results in one of the following $3$ cycles:

$53 \, 955 \to 59 \, 994 \to 53 \, 955$
$61 \, 974 \to 82 \, 962 \to 75 \, 933 \to 63 \, 954 \to 61 \, 974$
$62 \, 964 \to 71 \, 973 \to 83 \, 952 \to 74 \, 943 \to 62 \, 964$


Proof

We have:

\(\ds 95 \, 553 - 35 \, 559\) \(=\) \(\ds 59 \, 994\)
\(\ds 99 \, 954 - 45 \, 999\) \(=\) \(\ds 53 \, 995\)


\(\ds 97 \, 641 - 14 \, 679\) \(=\) \(\ds 82 \, 962\)
\(\ds 98 \, 622 - 22 \, 689\) \(=\) \(\ds 75 \, 933\)
\(\ds 97 \, 533 - 33 \, 579\) \(=\) \(\ds 63 \, 954\)
\(\ds 96 \, 543 - 34 \, 569\) \(=\) \(\ds 61 \, 974\)


\(\ds 96 \, 642 - 24 \, 669\) \(=\) \(\ds 71 \, 973\)
\(\ds 97 \, 731 - 13 \, 779\) \(=\) \(\ds 83 \, 952\)
\(\ds 98 \, 532 - 23 \, 589\) \(=\) \(\ds 74 \, 943\)
\(\ds 97 \, 443 - 34 \, 479\) \(=\) \(\ds 62 \, 964\)




Sources