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Question Number 8026 by Nayon last updated on 28/Sep/16

Find the factor of (3^(200) +4)

$${Find}\:{the}\:{factor}\:{of}\:\left(\mathrm{3}^{\mathrm{200}} +\mathrm{4}\right) \\ $$

Answered by Rasheed Soomro last updated on 28/Sep/16

3^(200) +4 =(3^(100) )^2 +2^2 =(3^(100) )^2 +2(3^(100) )(2)+2^2 −2(3^(100) )(2) =(3^(100) +2)^2 −(2.3^(50) )^2 =(3^(100) +2−2.3^(50) )(3^(100) +2+2.3^(50) ) =(3^(100) −2.3^(50) +2)(3^(100) +2.3^(50) +2)

$$\mathrm{3}^{\mathrm{200}} +\mathrm{4} \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} \right)^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} \right)^{\mathrm{2}} +\mathrm{2}\left(\mathrm{3}^{\mathrm{100}} \right)\left(\mathrm{2}\right)+\mathrm{2}^{\mathrm{2}} −\mathrm{2}\left(\mathrm{3}^{\mathrm{100}} \right)\left(\mathrm{2}\right) \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}\right)^{\mathrm{2}} −\left(\mathrm{2}.\mathrm{3}^{\mathrm{50}} \right)^{\mathrm{2}} \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}−\mathrm{2}.\mathrm{3}^{\mathrm{50}} \right)\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}+\mathrm{2}.\mathrm{3}^{\mathrm{50}} \right) \\ $$$$=\left(\mathrm{3}^{\mathrm{100}} −\mathrm{2}.\mathrm{3}^{\mathrm{50}} +\mathrm{2}\right)\left(\mathrm{3}^{\mathrm{100}} +\mathrm{2}.\mathrm{3}^{\mathrm{50}} +\mathrm{2}\right) \\ $$

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