14383
14383 (Gheorghe Gherasim)
Numerele naturale distincte a, b verifică 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 9 \cdot [\,a, b]\,=a \cdot b \cdot (\,a \cdot b)\,} .
i) Arătați că a și b nu sunt prime între ele.
ii) Arătați că diferența numerelor este cel puțin 3.
([a, b] reprezintă cel mai mic multiplu comun al numerelor a și b, iar (a, b) este cel mai mare divizor comun al numerelor a și b).
Soluție.
i) Se știe că 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 \cdot b= [\,a, b]\, \cdot (\,a, b)\,} și relația devine . De aici obținem 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, b)\,\}}^2=9} , de unde 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, b)\,=3} , ceea ce arată că a și b nu sunt prime între ele.
![{\displaystyle 9\cdot [\,a,b]\,=[\,a,b]\,\cdot {\{(\,a,b)\,\}}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9d7612a16693ddf9f422e48bf7654c2792da01)
ii) Din 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, b)\,=3} 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 a=3x} și 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 b=3y} 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 (\,x, y)\,=1} . Deoarece 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<b} avem 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<y} . Cum x și y sunt numere naturale avem 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 y-x \geq 1} . 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 b-a=3(\,y-x)\, \geq 3 \cdot 1=3} , de unde 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 b \geq a+3} .