Number Theory

How many zeros are at the end of $(100!)(200!)(300!)(400!)$ when multiplied out?

$\begin{array}{|rcll|} \hline \text{zeros} &=& \small{ \lfloor\frac{100}{5}\rfloor +\lfloor\frac{100}{25}\rfloor +\lfloor\frac{200}{5}\rfloor +\lfloor\frac{200}{25}\rfloor +\lfloor\frac{200}{125}\rfloor +\lfloor\frac{300}{5}\rfloor +\lfloor\frac{300}{25}\rfloor +\lfloor\frac{300}{125}\rfloor +\lfloor\frac{400}{5}\rfloor +\lfloor\frac{400}{25}\rfloor +\lfloor\frac{400}{125}\rfloor } \\\\ &=& 20+4+40+8+1+60+12+2+80+16+3 \\ \mathbf{\text{zeros}} &=& \mathbf{246} \\ \hline \end{array}$