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Omni-MATH / For an arbitrary positive integer m, not divisible by 3, consider the permutation x mapsto 3x pmod{m} on the set { 1,2,dotsc ,m-1}. This permutation can be decomposed into disjoint…

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problem

For an arbitrary positive integer mm, not divisible by 33, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for m=10m=10 the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers mm is the number of cycles odd?
Plain-text mathematical notation (without MathML)
For an arbitrary positive integer m, not divisible by 3, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for m=10 the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers m is the number of cycles odd?
Original LaTeX notation
For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\mapsto 2)$ and $(5\mapsto 5)$. For which integers $m$ is the number of cycles odd?

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Source and history

task_id
88e3f96c-5c29-5f64-8e72-e05c68bfde2f
task_key
test--88e3f96c-5c29-5f64-8e72-e05c68bfde2f
task_revision_id
1
config / split
/ test

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