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# Number Theory

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

Jul 31, 2021

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

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

Jul 31, 2021