MathDB

Let

N

be a positive integer. Initially, a positive integer

A

is written on the board. At each step, we can perform one of the following two operations with the number written on the board:
(i) Add

N

to the number written on the board and replace that number with the sum obtained;
(ii) If the number on the board is greater than

1

and has at least one digit

1

, then we can remove the digit

1

from that number, and replace the number initially written with this one (with removal of possible leading zeros)
For example, if

N = 63

and

A = 25

, we can do the following sequence of operations:

25 \rightarrow 88 \rightarrow 151 \rightarrow 51 \rightarrow 5

And if

N = 143

and

A = 2

, we can do the following sequence of operations:

2 \rightarrow 145 \rightarrow 288 \rightarrow 431 \rightarrow 574 \rightarrow 717 \rightarrow 860 \rightarrow 1003 \rightarrow 3

For what values of

N

is it always possible, regardless of the initial value of

A

on the blackboard, to obtain the number

1

on the blackboard, through a finite number of operations?

Problem Statement