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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $169$