Question Number 147661 by mathdanisur last updated on 22/Jul/21
$$\mathrm{41}^{\mathrm{3}} \:+\:\mathrm{42}^{\mathrm{3}} \:+\:\mathrm{43}^{\mathrm{3}} \:+\:...\:+\:\mathrm{59}^{\mathrm{3}} \\ $$$${Find}\:{the}\:{last}\:{three}\:{digits}\:{of}\:{the} \\ $$$${number} \\ $$
Answered by nimnim last updated on 22/Jul/21
$${S}=\left(\mathrm{1}^{\mathrm{3}} +\mathrm{2}^{\mathrm{3}} +\mathrm{3}^{\mathrm{3}} +....+\mathrm{59}^{\mathrm{3}} \right)−\left(\mathrm{1}^{\mathrm{3}} +\mathrm{2}^{\mathrm{3}} +\mathrm{3}^{\mathrm{3}} +....+\mathrm{40}^{\mathrm{3}} \right) \\ $$$$\:\:\:=\left(\frac{\mathrm{59}×\mathrm{60}}{\mathrm{2}}\right)^{\mathrm{2}} −\left(\frac{\mathrm{40}×\mathrm{41}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\:\:\:=\left(\cancel{\mathrm{17}70}\right)^{\mathrm{2}} −\left(\cancel{\mathrm{8}20}\right)^{\mathrm{2}} \\ $$$$\:\:\:=\left(..\mathrm{70}−..\mathrm{20}\right)\left(..\mathrm{70}+..\mathrm{20}\right)\:\rightarrow\:{last}\:{two}\:{digits} \\ $$$$\:\:\:=..\mathrm{50}×..\mathrm{90} \\ $$$$\:\:\:=....\mathrm{500}\rightarrow{last}\:{three}\:{digits} \\ $$
Commented by mathdanisur last updated on 23/Jul/21
$${Sir},\:\left(\frac{\mathrm{59}\centerdot\mathrm{60}}{\mathrm{2}}\right)−...\:{can}\:{you}\:{explain}\:{this} \\ $$$${part}\:{please}.?\:{how}\:{did}\:{you}\:{get}\:{it}.? \\ $$
Commented by mathdanisur last updated on 22/Jul/21
$${thanks}\:{Sir} \\ $$
Commented by mathdanisur last updated on 23/Jul/21
$$\left(\frac{\mathrm{59}\centerdot\mathrm{60}}{\mathrm{2}}\right)^{\mathrm{2}} −... \\ $$