Pandigital Integers remaining Pandigital on Multiplication

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Theorem

Certain pandigital integers remain pandigital when multiplying them by certain single-digit integers:

\(\ds 1 \, 098 \, 765 \, 432 \times 1\) \(=\) \(\ds 1 \, 098 \, 765 \, 432\) which is pandigital
\(\ds 1 \, 098 \, 765 \, 432 \times 2\) \(=\) \(\ds 2 \, 197 \, 530 \, 864\) which is pandigital
\(\ds 1 \, 098 \, 765 \, 432 \times 3\) \(=\) \(\ds 3 \, 296 \, 296 \, 296\)
\(\ds 1 \, 098 \, 765 \, 432 \times 4\) \(=\) \(\ds 4 \, 395 \, 061 \, 728\) which is pandigital
\(\ds 1 \, 098 \, 765 \, 432 \times 5\) \(=\) \(\ds 5 \, 493 \, 827 \, 160\) which is pandigital
\(\ds 1 \, 098 \, 765 \, 432 \times 6\) \(=\) \(\ds 6 \, 592 \, 592 \, 592\)
\(\ds 1 \, 098 \, 765 \, 432 \times 7\) \(=\) \(\ds 7 \, 691 \, 358 \, 024\) which is pandigital
\(\ds 1 \, 098 \, 765 \, 432 \times 8\) \(=\) \(\ds 8 \, 790 \, 123 \, 456\) which is pandigital
\(\ds 1 \, 098 \, 765 \, 432 \times 9\) \(=\) \(\ds 9 \, 888 \, 888 \, 888\)


The sequence:

$1039675824, 1053826974, 1068253974, 1068379524, 1073968254, 1075396824, 1098765432, 1204756839, 1234567890, 1357802469$

contains all pandigital integers with at least $4$ nontrivial pandigital multiples, of which:

$1098765432, 1234567890$

has $5$.

This sequence is A167476 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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