Sum of Factorials of Digits of 169

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Theorem

Let the factorials of the digits of $169$ be added.

Let the same process be done on the result.

Repeat.

After $3$ iterations, the result will be $169$.


Proof

\(\ds 1! + 6! + 9!\) \(=\) \(\ds 1 + 720 + 362 \, 880\)
\(\ds \) \(=\) \(\ds 363 \, 601\)
\(\ds 3! + 6! + 3! + 6! + 0! + 1!\) \(=\) \(\ds 6 + 720 + 6 + 720 + 1 + 1\)
\(\ds \) \(=\) \(\ds 1454\)
\(\ds 1! + 4! + 5! + 4!\) \(=\) \(\ds 1 + 24 + 120 + 24\)
\(\ds \) \(=\) \(\ds 169\)

$\blacksquare$


Sources