E:5743

E:5743 (Grigore Balog)

Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} mulțimea numerelor de forma Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{7a8b}} , care se divid cu Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 15} și $B$ mulțimea numerelor de forma ${\overline {7x8y}}$, care se divid cu $40$. Să se determine mulțimile $A\cup B$, $A\cap B$ și $A\setminus B$.

Soluție:

Dacă $15|{\overline {7a8b}}$, atunci $5|{\overline {7a8b}}$ și $3|{\overline {7a8b}}$. Din $5|{\overline {7a8b}}$, se obține $b\in \left\{0,5\right\}$. Din $3|{\overline {7a8b}}$, se obține $a+b\in {\mathcal {M}}_{3}$. Pentru $b=0$ rezultă $a\in \left\{0,3,6,9\right\}$. Pentru $b=5$ rezultă $a\in \left\{1,4,7\right\}$. Deci

$$A=\left\{n\in \mathbb {N} ,n={\overline {7a8b}},15|n\right\}=\left\{7080,7185,7380,7485,7680,7785,7980\right\}.$$

Dacă $40|{\overline {7x8y}}$, atunci $8|{\overline {7x8y}}$ și $5|{\overline {7x8y}}$. Cum $y$ este par, din $5|{\overline {7x8y}}$ se obține $y=0$. Rezultă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{7x80} \, \vdots \, 8} , ceea ce implică Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x \, \vdots \, 8} . Atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in \left\{0,2, 4, 6, 8\right\}} . DeciFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \left\{ n\in \mathbb{N}, n=\overline{7x8y} , 40 | n \right\} = \left\{ 7080, 7280, 7480, 7680, 7880\right\}.}