Question

Demonstrati ca numarul A=63ⁿ+7 la puterea n+1 x 3 la puterea 2n+1-21ⁿ x 3 la puterea n+2 , n € (apartine)N este divizibil cu 13

Answer 1

[tex]63^n+7^{n+1}\cdot3^{2n+1}-21^n\cdot3^{n+2}=\\ 63^n+7\cdot3\cdot7^n\cdot3^{2n}-3^2\cdot21^n \cdot3^n=\\ 63^n+7\cdot3\cdot7^n\cdot9^{n}-3^2\cdot21^n \cdot3^n=\\ 63^n+21\cdot63^n-9\cdot63^n=63^n(1+21-9)=13\cdot63^n\ \vdots\ 13 [/tex]