Question

Aratati ca oricare ar fi numerele naturale pentru care 2x - 3y = 4 , numarul (x-2)(y+2) este divizibil cu 6.

Answer 1

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Answer 2

[tex]\it 2x-3y=4 \Rightarrow x=\dfrac{4+3y}{2} \Rightarrow x = \dfrac{4+2y+y}{2} \Rightarrow x = 2+y+\dfrac{y}{2} \ \ \ \ (*) \\ \\ \\ x, \ y \in \mathbb{N} \stackrel{(*)}{\Longrightarrow} \dfrac{y}{2} \in\mathbb{N} \Rightarrow y=2k,\ \ k\in \mathbb{N} \\ \\ \\ y=2k \stackrel{(*)}{\Longrightarrow} x = 2+2k+k \Rightarrow x = 3k+2, k\in \mathbb{N} [/tex]

[tex]\it Pentru\ \ x = 3k+2,\ \ y=2k, \ avem: \\ \\ \\ (x-2)(y+2) = (3k+2-2)(2k+2)= 3k\cdot2(k+1) =6k(k+1) \Rightarrow \\ \\ \\ \Rightarrow 6\vdots (x-2)(y+2) [/tex]