Question

Va rog ajutor la astea 2 exerciții

Answer 1

$1)~Sa~presupunem~ca~cele~doua~numere~consecutive~au~un~divizor~comun~d.\\ \\ Avem~ca~\begin{cases}~d~|~n~\\d~~|~n+1~\end{cases} \\~\\ \xrightarrow{\text{daca~scazi~vei~avea~ca}}~~d~divide~1.\\ \\ Deci,~cele~doua~numere~consecutive~sunt~prime~intre~ele!\\ \\ \\ 2)~a=4n+5\\ \\~~~~ b=3n+4 \\ \\ La~fel,~sa~spunem~ca~cele~doua~numere~au~un~divizor~comun~d.\\ \\ \begin{cases}~d~|~4n+5~~|*3\\~d~|~3n+4~~|*4\end{cases} \Leftrightarrow~\begin{cases}~d~|~12n+15\\~d~|~12n+16~\end{cases}~\Leftrightarrow d~|~1\\ \\~Astfel,~cele~doua~numere~sunt~prime~intre~ele~\forall~n\in~N.$