Question Number 125959 by bramlexs22 last updated on 15/Dec/20
Answered by liberty last updated on 15/Dec/20
$${Let}\:{q}\:{denote}\:{the}\:{numbers}\:{of}\:{eggs}\:{originally} \\ $$$${in}\:{the}\:{basket}.\:{We}\:{want}\:{to}\:{find}\:{the}\:{least}\:{number} \\ $$$${of}\:{eggs}\:{in}\:{the}\:{basket}.\: \\ $$$$\Leftrightarrow\:{q}\:\equiv\:\mathrm{1}\left({mod}\:\mathrm{2}\right)\:\equiv\:\mathrm{1}\left({mod}\:\mathrm{3}\right)\equiv\:\mathrm{1}\left({mod}\:\mathrm{5}\right)\:\equiv\:\mathrm{0}\:\left({mod}\:\mathrm{7}\right) \\ $$$${the}\:{first}\:{three}\:{congruences}\:{then} \\ $$$${q}\:\equiv\:\mathrm{1}\:\left({mod}\:{lcm}\left(\mathrm{2},\mathrm{3},\mathrm{5}\right)\right)=\:\mathrm{30} \\ $$$${solving}\:{this}\:{with}\:{the}\:{remaining}\:{congruence} \\ $$$${gives}\:{us}\:{q}\:\equiv\:\mathrm{91}\:\left({mod}\:\mathrm{210}\right)\:.\:{Thus}\:{the} \\ $$$${least}\:{number}\:{of}\:{eggs}\:{is}\:\mathrm{91}. \\ $$$$ \\ $$