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OlympiadBench / 1964 / Let Z_(>0) denote the set of positive integers. For any positive…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Let

\mathbb{Z}_{>0}

denote the set of positive integers. For any positive integer

k

, a function

f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}

is called

k

-good if

\operatorname{gcd}(f(m)+n, f(n)+m) \leqslant k

for all

m \neq n

. Find all

k

such that there exists a

k

-good function.

Plain-text mathematical notation (without MathML)

Let Z_(>0) denote the set of positive integers. For any positive integer k, a function f:Z_(>0)→Z_(>0) is called k-good if gcd(f(m)+n,f(n)+m)≤k for all m≠n. Find all k such that there exists a k-good function.

Original LaTeX notation

Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ is called $k$-good if $\operatorname{gcd}(f(m)+n, f(n)+m) \leqslant k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.

answer type

Expression

is multiple answer

false

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID: 1964
task_id: fb6d36e2-97b6-59d6-851e-a5f3955744ae
task_key: OE~5fTO~5fmaths~5fen~5fCOMP--train--1964
task_revision_id: 3
config / split: OE_TO_maths_en_COMP / train

Official source

initial import