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# need help

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If m is the smallest positive integer such that m! is a multiple of 4125, and n is the smallest positive integer such that n! is a multiple of 2816000, then find n-m.

Jun 16, 2021

I claim that the $n=m=15$, so $n-m=0$.  To see that $15!$ is a common multiple of 4125 and 2816000, just check that $15!$ is a multiple of both $4125=3\cdot 5^3\cdot 11$  and $2816000=2^{11}\cdot 5^3\cdot 11$.  To see that $15!$ is the smallest common multiple of 4125 and 2816000, note that $14!$ is not a common multiple of 4125 and 2816000 (because $5^3$ does not divide $14!$ so it can't divide any $x!$ where $x \le 14$).