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Harvard-MIT Mathematics Tournament

Number theory Difficulty 4.6 AIME Prove it United States

Problem:
Find the largest number nn such that (2004!)!(2004!)! is divisible by ((n!)!)!((n!)!)!.

Solution

Solution:
For positive integers a,ba, b, we have
a!∣b!⇔a!≤b!⇔a≤b. a!\mid b!\quad \Leftrightarrow \quad a!\leq b!\quad \Leftrightarrow \quad a \leq b .
Thus,
((n!)!)!∣(2004!)!⇔(n!)!≤2004!⇔n!≤2004⇔n≤6. ((n!)!)!\mid(2004!)!\Leftrightarrow(n!)!\leq 2004!\Leftrightarrow n!\leq 2004 \quad \Leftrightarrow \quad n \leq 6 .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.