A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. Proposed by Carl Schildkraut and Holden Mui
Problem
Source: 2019 ELMO Shortlist N4
Tags: number theory, Divisibility
04.07.2019 17:46
oops please ignore
29.02.2020 16:22
What does $n|(a_0a_1...a_n)_b$ mean?
29.02.2020 16:30
It means $n$ divides the number with digits $a_0...a_n$ written in base $b$
14.08.2020 11:46
stroller wrote:
I dont understand this Can anyone solve it?
20.09.2021 10:37
pieater314159 wrote: A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$. Proposed by Carl Schildkraut and Holden Mui
01.02.2023 08:21
I wonder why we can get x0<x1<... the finite chain