Generalising Luke's problem

I have just had a wonderful week walking in the area of Agen in southwest France – with some time to relax. Spurred on by my energetic and delightful sister, Brenda, I have been reflecting on how to generalise Luke's problem (see my two previous blogs). This is what I have come up with.

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.