MathDB

An economist and a statistician play a game on a calculator which does only one operation. The calculator displays only positive integers and it is used in the following way: Denote by

n

an integer that is shown on the calculator. A person types an integer,

m

, chosen from the set

\{ 1, 2, . . . , 99 \}

of the first

99

positive integers, and if

m\%

of the number

n

is again a positive integer, then the calculator displays

m\%

n

. Otherwise, the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation looses. How many numbers from

\{1, 2, . . . , 2019\}

guarantee the winning strategy for the statistician, who plays second?
For example, if the calculator displays

1200

, the economist can type

50

, giving the number

600

on the calculator, then the statistician can type

25

giving the number

150

. Now, for instance, the economist cannot type

75

75\%

150

is not a positive integer, but can choose

40

and the game continues until one of them cannot type an allowed number
Proposed by Serbia

Problem Statement