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Omni-MATH / Let N_(≥1) be the set of positive integers.

Problem

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problem

Let N≥1\mathbb{N}_{\geqslant 1} be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers mm and nn: GCD(f(m),n)+LCM(m,f(n))=GCD(m,f(n))+LCM(f(m),n).\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) = \mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right). Note: if aa and bb are positive integers, GCD(a,b)\mathrm{GCD}(a,b) is the largest positive integer that divides both aa and bb, and LCM(a,b)\mathrm{LCM}(a,b) is the smallest positive integer that is a multiple of both aa and bb.
Plain-text mathematical notation (without MathML)
Let N_(≥1) be the set of positive integers. 
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers m and n:
GCD(f(m),n)+LCM(m,f(n))=GCD(m,f(n))+LCM(f(m),n).

Note: if a and b are positive integers, GCD(a,b) is the largest positive integer that divides both a and b, and LCM(a,b) is the smallest positive integer that is a multiple of both a and b.
Original LaTeX notation
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers. 
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]

Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.

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Source and history

task_id
28aa594b-f216-59fa-98de-c07c8624e0be
task_key
test--28aa594b-f216-59fa-98de-c07c8624e0be
task_revision_id
1
config / split
/ test

Official source

initial import