Question

Demonstrati ca A= 6^n+1 + 2^n × 3^n+1 + 2^n+2 × 3^n se divide cu 13 pentru orice n numar natural. va rog !!!

Answer 1

$A=6^{n+1}+2^n\cdot3^{n+1}+2^{n+2}\cdot3^n=6\cdot6^n+2^n\cdot3\cdot3^n+2^2\cdot2^n\cdot3^n=\\=6\cdot6^n+3\cdot(2\cdot3)^n+4\cdot(2\cdot3)^n=6\cdot6^n+3\cdot6^n+4\cdot6^n=13\cdot6^n$

care se divide cu 13.