Question

aratati ca :
5ⁿ x 3ⁿ + 5ⁿ⁺¹ x 3ⁿ⁺¹ +5ⁿ⁺¹x 3ⁿ⁺²-2 x 5ⁿ⁺² x 3ⁿ este divizil cu 11

Answer 1

$5^n*3^n+5^{n+1}*3^{n+1}+5^{n+1}*3^{n+2}-2*5^{n+2}*3^n=$

$(5*3)^n+5^n*5*3^n*3+5^n*5*3^n*3^2-2*5^n*5^2*3^n=$

$15^n+(5*3)^n*15+(5*3)^n*45-(5*3)^n*50=$

$15^n+15^n*15+15^n*45-15^n*50=$

$15^n(1+15+45-50)=$

$15^n*11=\textgreater 11|15^n*11=\textgreater11|5^n*3^n+5^{n+1}*3^{n+1}+5^{n+1}*3^{n+2}-2*5^{n+2}*3^n$