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Omni-MATH / Let a>1 be a positive integer and d>1 be a positive integer coprime to a. Let x₁=1, and for k≥1, define …

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problem

Let a>1a > 1 be a positive integer and d>1d > 1 be a positive integer coprime to aa. Let x1=1x_1=1, and for k≥1k\geq 1, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of aa and dd, the greatest positive integer nn for which there exists an index kk such that xkx_k is divisible by ana^n.
Plain-text mathematical notation (without MathML)
Let a>1 be a positive integer and d>1 be a positive integer coprime to a. Let x₁=1, and for k≥1, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of a and d, the greatest positive integer n for which there exists an index k such that x_(k) is divisible by a^(n).
Original LaTeX notation
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

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Source and history

task_id
1aac4b7c-bc92-51e8-bbb9-a579d8a42d02
task_key
test--1aac4b7c-bc92-51e8-bbb9-a579d8a42d02
task_revision_id
1
config / split
/ test

Official source

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