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Omni-MATH / For positive integers a and b such that a is coprime to b, define ord_(b)(a) as the least positive integer k such that b∣a^(k)−1, and define φ(a) to be the number of positive integ…

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problem

For positive integers aa and bb such that aa is coprime to bb, define ordb(a)\operatorname{ord}_{b}(a) as the least positive integer kk such that b∣ak−1b \mid a^{k}-1, and define φ(a)\varphi(a) to be the number of positive integers less than or equal to aa which are coprime to aa. Find the least positive integer nn such that ordn(m)<φ(n)10\operatorname{ord}_{n}(m)<\frac{\varphi(n)}{10} for all positive integers mm coprime to nn.
Plain-text mathematical notation (without MathML)
For positive integers a and b such that a is coprime to b, define ord_(b)(a) as the least positive integer k such that b∣a^(k)−1, and define φ(a) to be the number of positive integers less than or equal to a which are coprime to a. Find the least positive integer n such that ord_(n)(m)<(φ(n))/(10) for all positive integers m coprime to n.
Original LaTeX notation
For positive integers $a$ and $b$ such that $a$ is coprime to $b$, define $\operatorname{ord}_{b}(a)$ as the least positive integer $k$ such that $b \mid a^{k}-1$, and define $\varphi(a)$ to be the number of positive integers less than or equal to $a$ which are coprime to $a$. Find the least positive integer $n$ such that $$\operatorname{ord}_{n}(m)<\frac{\varphi(n)}{10}$$ for all positive integers $m$ coprime to $n$.

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task_id
6d76bcdb-3546-52c7-8569-2a874248049c
task_key
test--6d76bcdb-3546-52c7-8569-2a874248049c
task_revision_id
1
config / split
/ test

Official source

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