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mods with a twist

Source: EGMO 2023/5

April 16, 2023

EGMOEGMO 2023number theory

Problem Statement

We are given a positive integer

s \ge 2

. For each positive integer

k

, we define its twist

k’

as follows: write

k

as

as+b

, where

a, b

are non-negative integers and

b < s

, then

k’ = bs+a

. For the positive integer

n

, consider the infinite sequence

d_1, d_2, \dots

where

i

and

s^2-1

is the twist of

d_i

for each positive integer

i

. Prove that this sequence contains

1

if and only if the remainder when

n

is divided by

s^2-1

is either

1

or

s

.

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