Formulă.
Numărul de zerouri pentru n! în baza 10 este:
[tex]\displaystyle \sum\limits_{k=1}^{\infty}\Big[\dfrac{n}{5^k}\Big] = \Big[\dfrac{n}{5}\Big] +\Big[\dfrac{n}{5^2}\Big]+\Big[\dfrac{n}{5^3}\Big]+\Big[\dfrac{n}{5^4}\Big]+...[/tex]
[tex]10^9\,\big|\, (1\cdot 2\cdot 3\cdot...\cdot 40)\\ \\ \Big[\dfrac{40}{5^1}\Big] =8 \\ \\ \Big[\dfrac{40}{5^2}\Big] =1 \\ \\ \Big[\dfrac{40}{5^3}\Big]=0\\ \\ ................\\ \\ \Rightarrow 1\cdot 2\cdot 3\cdot...\cdot 40 \text{ are }8+1+0 = 9 \text{ zerouri} \\ \\ \Rightarrow 10^9\,\big|\, (1\cdot 2\cdot 3\cdot...\cdot 40)\quad \mathrm{q.e.d.}[/tex]
