MathDB

Grant is standing at the beginning of a hallway with infinitely many lockers, numbered,

1, 2, 3, \ldots

All of the lockers are initially closed. Initially, he has some set

S = \{1, 2, 3, \ldots\}

.
Every step, for each element

s

S

, Grant goes through the hallway and opens each locker divisible by

s

that is closed, and closes each locker divisible by

s

that is open. Once he does this for all

s

, he then replaces

S

with the set of labels of the currently open lockers, and then closes every door again.
After

2022

steps,

S

has

n

integers that divide

{10}^{2022}

. Find

n

.
Proposed by Oliver Hayman

Problem Statement