Properties of 12,345,679

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

Penholodigital Properties of 123,456,789

Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12,345,679$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12,345,679$