# Pandigital Properties of 987,654,321

Jump to navigation
Jump to search

## Theorem

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

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

\(\displaystyle 987 \, 654 \, 321 \times 1\) | \(=\) | \(\displaystyle 987 \, 654 \, 321\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 2\) | \(=\) | \(\displaystyle 1 \, 975 \, 308 \, 642\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 3\) | \(=\) | \(\displaystyle 2 \, 962 \, 962 \, 963\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 4\) | \(=\) | \(\displaystyle 3 \, 950 \, 617 \, 284\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 5\) | \(=\) | \(\displaystyle 4 \, 938 \, 271 \, 605\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 6\) | \(=\) | \(\displaystyle 5 \, 925 \, 925 \, 925\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 7\) | \(=\) | \(\displaystyle 6 \, 975 \, 308 \, 642\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 8\) | \(=\) | \(\displaystyle 7 \, 901 \, 234 \, 568\) | |||||||||||

\(\displaystyle 987 \, 654 \, 321 \times 9\) | \(=\) | \(\displaystyle 8 \, 888 \, 888 \, 889\) |

Also:

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

which is also pandigital.

## Sources

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