Question Number 482 by prakash jain last updated on 12/Jan/15
![The number 1000! has certain number of 0s at the end, what the the first non−zero digit. 1000!=...d_1 d_2 d_3 D00000... where d_1 ,d_2 ,d_3 ,D are digits. Find the value of digit D.](https://www.tinkutara.com/question/Q482.png)
$$\mathrm{The}\:\mathrm{number}\:\mathrm{1000}!\:\mathrm{has}\:\mathrm{certain}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{0}{s}\:\mathrm{at}\:\mathrm{the}\:\mathrm{end},\:\mathrm{what}\:\mathrm{the}\:\mathrm{the}\:\mathrm{first}\:\mathrm{non}−\mathrm{zero} \\ $$$$\mathrm{digit}. \\ $$$$\mathrm{1000}!=…\mathrm{d}_{\mathrm{1}} \mathrm{d}_{\mathrm{2}} \mathrm{d}_{\mathrm{3}} \mathrm{D00000}… \\ $$$$\mathrm{where}\:\mathrm{d}_{\mathrm{1}} ,\mathrm{d}_{\mathrm{2}} ,\mathrm{d}_{\mathrm{3}} ,\mathrm{D}\:\mathrm{are}\:\mathrm{digits}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{digit}\:\mathrm{D}. \\ $$
Commented by 123456 last updated on 12/Jan/15
![1000! 2×5=10 1000!≡0(mod 2^p )→max p 1000!≡0(mod 5^q )→max q 1000!≡0(mod 10^s )→max s≡min(max p,max q)](https://www.tinkutara.com/question/Q486.png)
$$\mathrm{1000}! \\ $$$$\mathrm{2}×\mathrm{5}=\mathrm{10} \\ $$$$\mathrm{1000}!\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{2}^{\mathrm{p}} \right)\rightarrow\mathrm{max}\:\mathrm{p} \\ $$$$\mathrm{1000}!\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{5}^{\mathrm{q}} \right)\rightarrow\mathrm{max}\:\mathrm{q} \\ $$$$\mathrm{1000}!\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{10}^{\mathrm{s}} \right)\rightarrow\mathrm{max}\:\mathrm{s}\equiv\mathrm{min}\left(\mathrm{max}\:\mathrm{p},\mathrm{max}\:\mathrm{q}\right) \\ $$