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Question Number 20907 by DKumar last updated on 07/Sep/17

$$Howmanyzeroes\left(0\right)therein1\times 2\times 3\times 4..........99\times 100$$

Answered by dioph last updated on 07/Sep/17

$$\mathrm{zeros}\mathrm{in}\mathrm{number}=\mathrm{maximum}\mathrm{integer}$$$$n\mathrm{such}\mathrm{that}{10}^n=2^n{5}^n\mathrm{divides}\mathrm{the}\mathrm{number}.$$$$\mathrm{As}\mathrm{there}\mathrm{is}\mathrm{plenty}\mathrm{more}\mathrm{factors}\mathrm{of}2\mathrm{than}$$$$\mathrm{factors}\mathrm{of}5,\mathrm{we}\mathrm{will}\mathrm{find}n=\mathrm{number}$$$$\mathrm{of}\mathrm{factors}\mathrm{of}5.$$$$\mathrm{There}\mathrm{is}\mathrm{one}\mathrm{for}\mathrm{each}\mathrm{multiple}\mathrm{of}5$$$$\mathrm{between}1\mathrm{and}100\left(20\mathrm{of}\mathrm{them}\right)+$$$$\mathrm{another}\mathrm{one}\mathrm{for}\mathrm{each}\mathrm{multiple}\mathrm{of}$$$$25\mathrm{between}1\mathrm{and}100\left(4\mathrm{of}\mathrm{them}\right).$$$$\mathrm{Hence},100!\mathrm{ends}\mathrm{in}24\mathrm{zeros}.$$

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