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Omni-MATH / An economist and a statistician play a game on a calculator which does only one

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problem

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 nn an integer that is shown on the calculator. A person types an integer, mm, chosen from the set {1,2,...,99}\{ 1, 2, . . . , 99 \} of the first 9999 positive integers, and if m%m\% of the number nn is again a positive integer, then the calculator displays m%m\% of nn. 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}\{1, 2, . . . , 2019\} guarantee the winning strategy for the statistician, who plays second? For example, if the calculator displays 12001200, the economist can type 5050, giving the number 600600 on the calculator, then the statistician can type 2525 giving the number 150150. Now, for instance, the economist cannot type 7575 as 75%75\% of 150150 is not a positive integer, but can choose 4040 and the game continues until one of them cannot type an allowed number
Plain-text mathematical notation (without MathML)
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% of 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 as 75% of 150 is not a positive integer, but can
choose 40 and the game continues until one of them cannot type an allowed number
Original LaTeX notation
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 $m\%$ of $n$. 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 $75$ as $75\%$ of $150$ is not a positive integer, but can
choose $40$ and the game continues until one of them cannot type an allowed number

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Source and history

task_id
c2d7aa14-7459-5586-b2b7-f46373f5bf0e
task_key
test--c2d7aa14-7459-5586-b2b7-f46373f5bf0e
task_revision_id
1
config / split
/ test

Official source

initial import