Question Number 106628 by deep last updated on 06/Aug/20
![(((p−1))/p) where p=prime no. Remainder will always be (p−1) or −1 Que. find Remainder ((1!+2!+3!+........................1000!)/(10)) Que. ((1!+2!+3!+........................1000!)/(12)) Que. ((1!+2!+3!+........................1000!)/9) Que. What id the unit digit of below expression 1!+2!+3!+4!+......................1000! ANS. If we divide some number by 100,then remainder is last 2digit similary 1000----Last 3digit 10000 last 4 digit 100000 last 5 digits [(((1+2+3+4+0+0+0+..........+0)/(10))),() ] R=3 unit digit =3](Q106628.png)
$$\frac{\left(\boldsymbol{{p}}−\mathrm{1}\right)}{\boldsymbol{{p}}}\:\:\:\:\:\:\boldsymbol{{where}}\:\boldsymbol{{p}}=\boldsymbol{{prime}}\:\boldsymbol{{no}}. \\ $$$$\boldsymbol{{R}}{emainder}\:{will}\:{always}\:{be}\:\left({p}−\mathrm{1}\right)\:{or}\:−\mathrm{1} \\ $$$$ \\ $$$$ \\ $$$${Que}.\:{find}\:{Remainder} \\ $$$$\frac{\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+........................\mathrm{1000}!}{\mathrm{10}} \\ $$$${Que}. \\ $$$$\frac{\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+........................\mathrm{1000}!}{\mathrm{12}} \\ $$$$\boldsymbol{{Q}}{ue}.\:\frac{\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+........................\mathrm{1000}!}{\mathrm{9}} \\ $$$$ \\ $$$${Que}.\:{What}\:{id}\:{the}\:{unit}\:{digit}\:{of}\:{below} \\ $$$${expression} \\ $$$$\mathrm{1}!+\mathrm{2}!+\mathrm{3}!+\mathrm{4}!+......................\mathrm{1000}! \\ $$$${ANS}.\:\:\:{If}\:{we}\:{divide}\:{some}\:{number}\:{by}\:\mathrm{100},{then}\:{remainder}\:{is}\:{last}\:\mathrm{2}{digit}\: \\ $$$$ \\ $$$${similary}\:\:\:\mathrm{1000}----{Last}\:\mathrm{3}{digit} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{10000}\:\:\:\:\:\:\:{last}\:\mathrm{4}\:{digit} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{100000}\:\:\:\:\:\:{last}\:\mathrm{5}\:{digits} \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\begin{bmatrix}{\frac{\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{0}+\mathrm{0}+\mathrm{0}+..........+\mathrm{0}}{\mathrm{10}}}\\{}\end{bmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{R}}=\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{{unit}}\:\boldsymbol{{digit}}\:=\mathrm{3} \\ $$$$ \\ $$