Biggest-prime-factor-of-3-14-3-13-12-

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

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

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

Biggest prime factor 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 {4 (3^{12} - 1)}

= 12 {{(3^{6})}^{2} - {(1)}^{2}}

= 12 (3^{6} - 1) (3^{6} + 1)

= 12 ({(3^{2})}^{3} - 1^{3}) {(3^{2})}^{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)

73 i s t h e b i g g e s t p r i m e

Commented by BaliramKumar last updated on 02/Dec/23

Nice solution

thanks

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

A nother W 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 {{(3^{2})}^{3} - 1^{3}} {{(3^{2})}^{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 73 i s t h e b i g g e s t p r i m e

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

A n o t h e r v e r s i o n

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

= 3^{14} - 9 + 3^{13} - 3

= 9 (3^{12} - 1) + 3 (3^{12} - 1)

= 12 (3^{12} - 1)

\dots .