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Question Number 201257 by BaliramKumar last updated on 02/Dec/23

$$\mathrm{Biggest}\mathrm{prime}\mathrm{factor}\mathrm{of}(3^{14}+3^{13}-12)=?$$

Answered by Rasheed.Sindhi last updated on 02/Dec/23

$$\mathrm{Biggest}\mathrm{prime}\mathrm{factor}\mathrm{of}(3^{14}+3^{13}-12)=?$$$$=3(3^{13}+3^{12}-4)$$$$=3(3^{13}-3+3^{12}-1)$$$$=3\{3(3^{12}-1)+(3^{12}-1)\}$$$$=3\left\{4(3^{12}-1)\right\}$$$$=12\{{\left(3^6\right)}^2-{\left(1\right)}^2\}$$$$=12(3^6-1)(3^6+1)$$$$=12({\left(3^2\right)}^3-1^3){\left(3^2\right)}^3+1^3)$$$$=12(3^2-1)(3^4+3^2+1)(3^2+1)(3^4-3^2+1)$$$$=12\left(8\right)\left(91\right)\left(10\right)\left(73\right)$$$$=12\left(8\right)\left(7\right)\left(13\right)\left(10\right)\left(73\right)$$$$73isthebiggestprime$$

Commented by BaliramKumar last updated on 02/Dec/23

$$\mathrm{Nice}\mathrm{solution}$$$$\mathrm{thanks}$$

Answered by Rasheed.Sindhi last updated on 02/Dec/23

$$\mathbb{A}\mathrm{nother}\mathbb{W}\mathrm{ay}$$$$3^{14}+3^{13}-12)$$$$=3^{13}(3+1)-12$$$$=4(3^{13}-3)$$$$=4.3(3^{12}-1)$$$$=12(3^6-1)(3^6+1)$$$$=12\{{\left(3^2\right)}^3-1^3\}\{{\left(3^2\right)}^3+1^3\}$$$$=12(3^2-1)(3^4+3^2+1)(3^2+1)(3^4-3^2+1)$$$$=12.8.91.10.73$$$$=12.8.7.13.10.73\Rightarrow 73isthebiggestprime$$

Commented by Rasheed.Sindhi last updated on 03/Dec/23

$$\mathcal{A}notherversion$$$$3^{14}+3^{13}-12$$$$=3^{14}-9+3^{13}-3$$$$=9(3^{12}-1)+3(3^{12}-1)$$$$=12(3^{12}-1)$$$$....$$

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