Question

Determinați numărul natural xy care verifică relația
$xy = 3x + 5y$

Answer 1

$xy = 3x+5y\\ \\ xy - 5y = 3x \\ \\ y(x-5) = 3x \\ \\ y = \dfrac{3x}{x-5} \\ \\ \dfrac{3x}{x-5} \in \mathbb{Z} \Rightarrow \dfrac{3x-15+15}{x-5} \in \mathbb{Z} \Rightarrow \dfrac{3(x-5)}{x-5} + \dfrac{15}{x-5} \in \mathbb{Z} \Rightarrow \\ \\ \Rightarrow 3 + \dfrac{15}{x-5} \in \mathbb{Z} \Rightarrow (x-5) ~|~ 15 \Rightarrow \\ \\ \Rightarrow (x-5) \in \Big\{-15,-5,-3,3,5,15\Big\} \Big|+5 \\ \\ \Rightarrow x \in \Big\{-10,0,2,8,10,20\}$

$\Rightarrow \boxed{x \in \Big\{2,8\Big\}}\Big|-5 \\ \\ \Rightarrow (x-5)\in \Big\{-3,3\Big\} \Big|^{-1} \Rightarrow \dfrac{1}{x-5} \in \Big\{-\dfrac{1}{3}, \dfrac{1}{3}\Big\} \Big|\cdot 15 \Rightarrow \\ \\ \Rightarrow \dfrac{15}{x-5} \in \Big\{-5,5\Big\} \Big|+3 \Rightarrow \left(3 + \dfrac{15}{x-5} \right) \in \Big\{-2,8\Big\} \Rightarrow \\ \\ \Rightarrow y \in \Big\{-2,8\Big\} \Rightarrow \boxed{y \in \Big\{\emptyset,8\Big\}}$

$\Rightarrow (x,y) \in \Big\{(8,8)\Big\} \Rightarrow xy = 8\cdot 8 \Rightarrow \boxed{xy = 64}$