Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Prove that $$(n!)\cdot(m!)^n|(mn)!$$ I can prove it using Legendre's Formula, but I have to use the lemma that $$ ...
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Clarification regarding Spivak, exercise 1-24

The 24th problem in the first chapter of Spivak's Calculus has to do with proving that the placement of parentheses in a sum is irrelevant. Let $s(a_1, a_2,... a_{n})$ denote some sum formed from ...