Question

Aratati ca numarul p=27 la puterea 17 - 81 la puterea 12 se imparte exact la 13.

Answer 1

   
[tex]\displaystyle\\ 27^{17}-81^{12} =\\\\ = \Big(3^3\Big)^{17}-\Big(3^4\Big)^{12}=\\\\ =3^{3\times 17}-3^{4\times 12}\\\\ =3^{51} -3^{48}=~~~~\text{Dam factor comun pe } 3^{48}\\\\ =3^{48}\Big(3^{51-48}-3^{48-48} \Big)=\\\\ =3^{48}\Big(3^{3}-3^{0} \Big)=3^{48}(27-1) = 26\times3^{48}= 13\times2\times3^{48}~\vdots~13 [/tex]