Question 10

If $$N = 3^{14} + 3^{13} - 12$$, then what is the largest prime factor of $$N$$?

Solution

$$3^{14}+3^{13}-12$$

$$=3^{13}\left(3+1\right)-12=3.4\left(3^{12}-1\right)$$

$$=3.4\left(3^6-1\right)\left(3^6+1\right)$$

$$=3.4\left(3^2-1\right)\left(3^4+3^2+1\right)\left(3^2+1\right)\left(3^4-3^2+1\right)$$

$$=2^6.3.5.7.13,73$$

Largest prime factor is 73

D is correct choice.

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