Question

Se considera numerele naturale nenule si distincte a si b pentru care $a+\frac{4}{a} =b+\frac{4}{b}$ . Aratati ca numarul n=3ab-10 este natural

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

 

$\displaystyle\\a\in N,~b\in N\\\\\text{Cautam valori pentru a si b printre divizorii naturali ai lui 4. }\\\\D_4=\{1;~2;~4\}\\\\a=1\implies~a+\frac{4}{a}=1+\frac{4}{1}=1+4=5\\\\b\neq a\\\\b=2\implies~b+\frac{4}{b}=2+\frac{4}{2}=2+2=4;~\text{solutie respinsa deoarece }~4\neq5\\\\b=4\implies~b+\frac{4}{b}=4+\frac{4}{4}=4+1=5\\\\\text{Rezulta solutia:}\\\\\boxed{\bf~a=1~si~b=4}\\n=3ab-10=3\cdot1\cdot4-10=12-10=\boxed{2\in N}$