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2025 6th Memorial "Aleksandar Blazhevski-Cane", P5
Tags: number theory , divisibility , 2-adic valuation
Let $s < t$ be positive integers. Define a sequence by: $a_1 = s, a_2 = t$; $a_3$ is the smallest integer that's greater than $a_2$ and divisible by $a_1$; in general, $a_{n + 1}$ is the smallest integer greater than $a_n$ that's divisible by $a_1, a_2, ..., a_{n - 2}, a_{n - 1}$. [b]a)[/b] What is the maximum number of odd integers that can appear in such a sequence? (Justify your answer) [b]b)[/b] Prove that $a_{2025}$ is divisible by $2^{808}$, regardless of the choice of $s$ and $t$. Proposed by [i]Ilija Jovcevski[/i]