Question Number 8026 by Nayon last updated on 28/Sep/16
$${Find}\:{the}\:{factor}\:{of}\:\left(\mathrm{3}^{\mathrm{200}} +\mathrm{4}\right) \\ $$
Answered by Rasheed Soomro last updated on 28/Sep/16
$$\mathrm{3}^{\mathrm{200}} +\mathrm{4} \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} \right)^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} \right)^{\mathrm{2}} +\mathrm{2}\left(\mathrm{3}^{\mathrm{100}} \right)\left(\mathrm{2}\right)+\mathrm{2}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{3}^{\mathrm{100}} \right)\left(\mathrm{2}\right) \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}\right)^{\mathrm{2}} −\left(\mathrm{2}.\mathrm{3}^{\mathrm{50}} \right)^{\mathrm{2}} \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}−\mathrm{2}.\mathrm{3}^{\mathrm{50}} \right)\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}+\mathrm{2}.\mathrm{3}^{\mathrm{50}} \right) \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} −\mathrm{2}.\mathrm{3}^{\mathrm{50}} +\mathrm{2}\right)\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}.\mathrm{3}^{\mathrm{50}} +\mathrm{2}\right) \\ $$