Properties of 12,345,679
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Theorem
$12 \, 345 \, 679$ has the following properties:
| \(\ds 12 \, 345 \, 679 \times 1\) | \(=\) | \(\ds 12 \, 345 \, 679\) | digit $8$ is missing | |||||||||||
| \(\ds 12 \, 345 \, 679 \times 2\) | \(=\) | \(\ds 24 \, 691 \, 358\) | digit $7$ is missing | |||||||||||
| \(\ds 12 \, 345 \, 679 \times 3\) | \(=\) | \(\ds 37 \, 037 \, 037\) | ||||||||||||
| \(\ds 12 \, 345 \, 679 \times 4\) | \(=\) | \(\ds 49 \, 382 \, 716\) | digit $5$ is missing | |||||||||||
| \(\ds 12 \, 345 \, 679 \times 5\) | \(=\) | \(\ds 61 \, 728 \, 395\) | digit $4$ is missing | |||||||||||
| \(\ds 12 \, 345 \, 679 \times 6\) | \(=\) | \(\ds 74 \, 074 \, 074\) | ||||||||||||
| \(\ds 12 \, 345 \, 679 \times 7\) | \(=\) | \(\ds 86 \, 419 \, 753\) | digit $2$ is missing | |||||||||||
| \(\ds 12 \, 345 \, 679 \times 8\) | \(=\) | \(\ds 98 \, 765 \, 432\) | digit $1$ is missing | |||||||||||
| \(\ds 12 \, 345 \, 679 \times 9\) | \(=\) | \(\ds 111 \, 111 \, 111\) |
In each product, the sequence $1$ to $9$, with the one given digit missing, can be read in order by cycling round it, skipping a fixed number of digits (counting an extra one when going from start to end), for example:
- $2 \ (4691) \ 3 \ (58?2) \ 4 \ (6913) \ 5 \ (8?24) \ 6 \ (9135) \ 8 (?246) \ 9$
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Also see
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12,345,679$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12,345,679$