Penholodigital Properties of 123,456,789
Jump to navigation
Jump to search
Theorem
$123 \, 456 \, 789$ has the following properties:
It is penholodigital, and remains so when multiplied by $2$, $4$, $5$, $7$ and $8$:
\(\ds 123 \, 456 \, 789 \times 1\) | \(=\) | \(\ds 123 \, 456 \, 789\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 2\) | \(=\) | \(\ds 246 \, 913 \, 578\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 3\) | \(=\) | \(\ds 370 \, 370 \, 367\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 4\) | \(=\) | \(\ds 493 \, 827 \, 156\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 5\) | \(=\) | \(\ds 617 \, 283 \, 945\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 6\) | \(=\) | \(\ds 740 \, 740 \, 734\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 7\) | \(=\) | \(\ds 864 \, 197 \, 523\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 8\) | \(=\) | \(\ds 987 \, 654 \, 312\) | ||||||||||||
\(\ds 123 \, 456 \, 789 \times 9\) | \(=\) | \(\ds 1 \, 111 \, 111 \, 101\) |
This article is complete as far as it goes, but it could do with expansion. In particular: Add some mathematical analysis explaining this phenomenon You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Examples
This property turns up every so often in books on recreational mathematics and general puzzles.
Example: $8$
- Place in a row $9$ digits each different from the others.
- Multiply them by $8$, and the product shall still consist of $9$ different digits.
Also see
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $123,456,789$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $123,456,789$