Pandigital Product of Pandigital Pairs in 3 Ways

Theorem

The pandigital integer $0 \, 429 \, 315 \, 678$ can be expressed as the product of a pandigital doubleton in $3$ different ways:

$\ds 0 \, 429 \, 315 \, 678$

$=$

$\ds 04 \, 926 \times 87 \, 153$

$\ds $

$=$

$\ds 07 \, 923 \times 54 \, 186$

$\ds $

$=$

$\ds 15 \, 846 \times 27 \, 093$

We have that:

$0 \, 429 \, 315 \, 678 = 2 \times 3^2 \times 11 \times 19 \times 139 \times 821$

Then:

$\ds 04 \, 926 \times 87 \, 153$

$=$

$\ds \paren {2 \times 3 \times 821} \times \paren {3 \times 11 \times 19 \times 139}$

$\ds 07 \, 923 \times 54 \, 186$

$=$

$\ds \paren {3 \times 19 \times 139} \times \paren {2 \times 3 \times 11 \times 821}$

$\ds 15 \, 846 \times 27 \, 093$

$=$

$\ds \paren {2 \times 3 \times 19 \times 139} \times \paren {3 \times 11 \times 821}$

$\blacksquare$

1973: André Gouffé: Products Using All Ten Digits in the Denary System (J. Recr. Math. Vol. 6, no. 1)