After writing a solution to this problem I focused on product of digits and got the following result. Also I have a challenge also for all those who found this interesting try if you can do it!
Let be defined as the product of digits of when written in base for example . Then,
First, we have to see the following lemma:
We can remove numbers from to ( ) as the numbers between have a in them
Similarly we have to remove numbers from to ( and 1 ), from to ( and 1 ) and so on.
We can take leading digits common, reducing a single digit from each number
Now we can take by include numbers like as so it makes no change
Using the lemma we can get
If you apply it again and again
- I have proved this for base you can try for any other base or you may prove it for any base