I want to know if there are any tricks or shortcuts to write the factorial of a large number, like $20!,$ as the product of its prime factors. For example, $5!= 5 \times 3 \times 2^3$
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$\begingroup$ without using calculators, because this question appears in an mathematics contest $\endgroup$ – Tom.J Dec 21 '17 at 4:13
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1$\begingroup$ Given $n!$, it is enough to know the factorizations of 1 to $n$, e.g., $2 = 2^1 \times 3^0 \times 5^0$, $3 = 2^0 \times 3^1 \times 5^0$, $4 = 2^2 \times 3^0 \times 5^0$, $5 = 2^0 \times 3^0 \times 5^1$, so then $5! = 2^3 \times 3^1 \times 5^1$. $\endgroup$ – Robert Soupe Dec 21 '17 at 5:04
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