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HMMT February

Number theory Difficulty 4.8 AIME Find the answer

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

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

For positive integers a,ba, b, we have a!∣b!⇔a!≤b!⇔a≤ba!\mid b!\quad \Leftrightarrow 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 \Leftrightarrow n \leq 6

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