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Distance of integers

Source: Baltic Way 2015

November 8, 2015

number theory

Problem Statement

For any integer

n \ge2

, we define

A_n

to be the number of positive integers

m

with the following property: the distance from

n

to the nearest multiple of

m

is equal to the distance from

n^3

to the nearest multiple of

m

. Find all integers

n \ge 2

for which

A_n

is odd. (Note: The distance between two integers

a

and

b

is defined as

|a -b|

.)

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