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
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $987,654,321$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $987,654,321$