Question

Determinati toate perechile (a,b) astfel incat a/7 = 2/2b+1

Answer 1

produsul mezilor=produsul extremilor
a(2b+1)=14
14=1×14,14×1
14=2×7,7×2
a=1
2b+1=14 nu convine nu e Nr natural
a=14
2b+1=1
b=0
(14,0)
a=2
2b+1=7
2b=6
b=3
(2,3)
a=7
2b+1=2 nu convine nu e Nr natural

Answer 2

[tex]\it \dfrac{a}{7} = \dfrac{2}{2b+1} \Rightarrow a(2b+1)=14 \Rightarrow a\in D_{14},\ \ \ 2b+1 \in D_{14} \ \ \ \ \ (1) \\\;\\ \\\;\\ 2b+1 =impar \ \stackrel{(1)}{\Longrightarrow } 2b+1 \in \{1,\ 7\}_{-1} \Rightarrow 2b \in \{0,\ 6\} |_{:2} \Rightarrow \\\;\\ \\\;\\ \Rightarrow b \in \{0,\ \ 3\} \ \ \ (*)[/tex]

[tex]\it\ 2b+1 \in \{1,\ 7\} \stackrel{(1)}\Longrightarrow } \begin{cases} \it \dfrac{a}{7} =\dfrac{2}{1} \Rightarrow a = \dfrac{7\cdot2}{1} = 14 \\\;\\ \\\;\\ \it \dfrac{a}{7} =\dfrac{2}{7} \Rightarrow a = \dfrac{7\cdot2}{7} = 2 \end{cases}[/tex]

$\it\ Deci,\ (a,\ b) \in \{ (2,\ 3), \ \ (14,\ 0)\}$