Question

Vă rog mult, dacă ştie cineva să rezolve exercitiul !

Answer 1

$n = 5^1+5^2+5^3+...+5^{2016} \\ \\ 780_{(10)} \text{ in baza 5 este 11110}_{(5)} \\\\ \Rightarrow 780= 625+125+25+5 = 5^1+5^2+5^3+5^4\\ \\\\ n = 5^0\cdot (5^1 + 5^2+5^3+5^4)+5^4\cdot (5^1 + 5^2+5^3+5^4) +...+...\\ \\ \text{Verificam daca }2016\text{ apartine sirului 0,4,8,12,16,..., adica,}\\\text{daca e multiplu de 4.}\\\\ 2016:4 = 504\text{ rest 0}\quad (A) \\ \\ n = 5^0\cdot (5^1 + 5^2+5^3+5^4)+5^4\cdot (5^1 + 5^2+5^3+5^4) +...+\\ +5^{2012}\cdot (5^{1}+5^{2}+5^{3}+5^{4}) \\ \\ n = 780\cdot (5^0+5^4+5^8+...+5^{2012})\\ \\ \Rightarrow n\, \vdots \, 780$