Suppose that we have two operations that we can perform on an integer:

Multiply it by any positive integer.

Delete the 0's in its decimal representation.

Beginning with any positive integer can we always obtain a single-digit number after a finite number of operations? For example, beginning with 7, we can multiply by 15 to obtain 105, delete the 0 to get 15, multiply by 2 to get 30, then delete the 0 to end with 3.

(In reply to

some efforts by Steven Lord)

Looking through your nuts I see that none of them (other than 11) are from either the repunits {11, 111, 1111, 11111, 111111, ...} or rep-7s {77, 777, 7777, 77777, 777777, ...}

We have a very old puzzle on the site

Niners. This puzzle can be easily modified to show that if n is coprime to 10 then there are numbers m1 and m7 such that n*m1 is a repunit ant n*m7 is a rep-7.

Then if we could prove all repunits or all rep-7s can be reduced to a single digit then it would follow that all numbers can be reduced as well.