MathDB

Let

n \geq 3

be a positive integer. In an election debate, we have

n

seats arranged in a circle and these seats are numbered from

1

n

, clockwise. In each of these chairs sits a politician, who can be a liar or an honest one. Lying politicians always tell lies, and honest politicians always tell the truth.
At one heated moment in the debate, they accused each other of being liars, with the politician in chair

1

saying that the politician immediately to his left is a liar, the politician in chair

2

saying that all the

2

politicians immediately to his left are liars, the politician in the char

3

saying that all the

3

politicians immediately to his left are liars, and so on. Note that the politician in chair

n

accuses all

n

politicians (including himself) of being liars.
For what values of

n

is this situation possible to happen?

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?

3

Problems(2)