Question Number 113336 by 175mohamed last updated on 12/Sep/20
$${If}\:\mathrm{1},\:{a}^{\mathrm{2}} ,{a}^{\mathrm{3}} \:,...,{a}^{{n}−\mathrm{1}} \:{are}\:{the}\:{roots}\: \\ $$$${nth}\:{of}\:{unity}\:,\: \\ $$$${prove}\:{that}\::\:\left(\mathrm{1}+{a}\right)\left(\mathrm{1}+{a}^{\mathrm{2}} \right)\left(\mathrm{1}+{a}^{\mathrm{3}} \right)...\left(\mathrm{1}+{a}^{{n}−\mathrm{1}} \right) \\ $$$$=\:{n}−\mathrm{2}\lfloor\frac{{n}}{\mathrm{2}}\rfloor \\ $$$$ \\ $$