There are no two- or three-digit solutions.
There are solutions for any other number of digits, but only for multiplying by 4 (I am not including the trivial solution of multiplying a number like 7777 by 1).
The solutions are as follows:
2178 × 4 = 8712
21978 × 4 = 87912
219978 × 4 = 879912
2199978 × 4 = 8799912
21999978 × 4 = 87999912
219999978 × 4 = 879999912
...
and so on, ad infinitum!
What a nice problem that proved to be! Thank you, grandson Luke.
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