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Omni-MATH / Call a rational number [i]short[/i] if it has finitely many digits in its…

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problem

Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer mm, we say that a positive integer tt is m−m-[i]tastic[/i] if there exists a number c∈{1,2,3,…,2017}c\in \{1,2,3,\ldots ,2017\} such that 10t−1c⋅m\dfrac{10^t-1}{c\cdot m} is short, and such that 10k−1c⋅m\dfrac{10^k-1}{c\cdot m} is not short for any 1≤k<t1\le k<t. Let S(m)S(m) be the set of m−m-tastic numbers. Consider S(m)S(m) for m=1,2,….m=1,2,\ldots{}. What is the maximum number of elements in S(m)S(m)?
Plain-text mathematical notation (without MathML)
Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer m, we say that a positive integer t is m−[i]tastic[/i] if there exists a number c∈{1,2,3,…,2017} such that (10^(t)−1)/(c⋅m) is short, and such that (10^(k)−1)/(c⋅m) is not short for any 1≤k<t. Let S(m) be the set of m−tastic numbers. Consider S(m) for m=1,2,…. What is the maximum number of elements in S(m)?
Original LaTeX notation
Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?

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task_id
6fd28484-7de1-5ee9-afb1-067f80120ba9
task_key
test--6fd28484-7de1-5ee9-afb1-067f80120ba9
task_revision_id
1
config / split
/ test

Official source

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