Penholodigital Properties of 987,654,321

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Theorem

$987 \, 654 \, 321$ has the following properties:

It is penholodigital, and remains so when multiplied by $1$, $2$, $4$, $5$, $7$ and $8$:

\(\ds 987 \, 654 \, 321 \times 1\) \(=\) \(\ds 987 \, 654 \, 321\)
\(\ds 987 \, 654 \, 321 \times 2\) \(=\) \(\ds 1 \, 975 \, 308 \, 642\)
\(\ds 987 \, 654 \, 321 \times 3\) \(=\) \(\ds 2 \, 962 \, 962 \, 963\)
\(\ds 987 \, 654 \, 321 \times 4\) \(=\) \(\ds 3 \, 950 \, 617 \, 284\)
\(\ds 987 \, 654 \, 321 \times 5\) \(=\) \(\ds 4 \, 938 \, 271 \, 605\)
\(\ds 987 \, 654 \, 321 \times 6\) \(=\) \(\ds 5 \, 925 \, 925 \, 925\)
\(\ds 987 \, 654 \, 321 \times 7\) \(=\) \(\ds 6 \, 975 \, 308 \, 642\)
\(\ds 987 \, 654 \, 321 \times 8\) \(=\) \(\ds 7 \, 901 \, 234 \, 568\)
\(\ds 987 \, 654 \, 321 \times 9\) \(=\) \(\ds 8 \, 888 \, 888 \, 889\)


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Also:

$987 \, 654 \, 321 - 123 \, 456 \, 789 = 864 \, 197 \, 532$

which is also penholodigital.


Sources

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