Penholodigital Properties of 123,456,789

From ProofWiki
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

Retrieved from "https://proofwiki.org/w/index.php?title=Penholodigital_Properties_of_123,456,789&oldid=552289"