Luke's challenge

Grandson Luke set Grandad (me) this intriguing little mathematical challenge.

Find a 5-digit number, which when multiplied by 4 gives a 5-digit result with the digits of the original number in the reverse order.

It took Grandad about 10 minutes to crack this (without a calculator to hand).

I then started thinking whether there are other problems like this ... a 5-digit number when multiplied by something other than 4 giving a 5-digit result with the digits reversed? A 4-digit number version? A 3-digit number version? Driving home in the car yesterday I convinced myself that there is not a 2-digit version.

Something to investigate further when I have a few minutes free to do something pointless ...