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Question Number 482 by prakash jain last updated on 12/Jan/15

$$\mathrm{The}\mathrm{number}1000!\mathrm{has}\mathrm{certain}\mathrm{number}$$$$\mathrm{of}0s\mathrm{at}\mathrm{the}\mathrm{end},\mathrm{what}\mathrm{the}\mathrm{the}\mathrm{first}\mathrm{non}-\mathrm{zero}$$$$\mathrm{digit}.$$$$1000!=...{\mathrm{d}}_1{\mathrm{d}}_2{\mathrm{d}}_3\mathrm{D}00000...$$$$\mathrm{where}{\mathrm{d}}_1,{\mathrm{d}}_2,{\mathrm{d}}_3,\mathrm{D}\mathrm{are}\mathrm{digits}.$$$$\mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{digit}\mathrm{D}.$$

Commented by 123456 last updated on 12/Jan/15

$$1000!$$$$2\times 5=10$$$$1000!\equiv 0\left(\mathrm{mod}{2}^{\mathrm{p}}\right)\to \max \mathrm{p}$$$$1000!\equiv 0\left(\mathrm{mod}{5}^{\mathrm{q}}\right)\to \max \mathrm{q}$$$$1000!\equiv 0\left(\mathrm{mod}{10}^{\mathrm{s}}\right)\to \max \mathrm{s}\equiv \min (\max \mathrm{p},\max \mathrm{q})$$

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