E:5743: Difference between revisions
No edit summary |
No edit summary |
||
| Line 3: | Line 3: | ||
''Fie <math>A</math> mulțimea numerelor de forma <math>\overline{7a8b}</math>, care se divid cu <math>15</math> și <math>B</math> mulțimea numerelor de forma <math>\overline{7x8y}</math>, care se divid cu <math>40</math>. Să se determine mulțimile <math>A \cup B</math>, <math>A\cap B</math> și <math>A\setminus B</math>.'' | ''Fie <math>A</math> mulțimea numerelor de forma <math>\overline{7a8b}</math>, care se divid cu <math>15</math> și <math>B</math> mulțimea numerelor de forma <math>\overline{7x8y}</math>, care se divid cu <math>40</math>. Să se determine mulțimile <math>A \cup B</math>, <math>A\cap B</math> și <math>A\setminus B</math>.'' | ||
'''Soluție:''' | |||
Dacă <math>15 | \overline{7a8b}</math>, atunci <math>5 | \overline{7a8b}</math> și <math>3 | \overline{7a8b}</math>. Din <math>5 | \overline{7a8b}</math>, se obține <math> b \in \left\{0,5\right\}</math>. Din <math>3 | \overline{7a8b}</math>, se obține <math> a+b \in \mathcal{M}_3</math>. Pentru <math>b=0</math> rezultă <math> a \in \left\{0,3,6,9\right\}</math>. Pentru <math>b=5</math> rezultă <math> a \in \left\{1,4,7\right\}</math>. Deci | Dacă <math>15 | \overline{7a8b}</math>, atunci <math>5 | \overline{7a8b}</math> și <math>3 | \overline{7a8b}</math>. Din <math>5 | \overline{7a8b}</math>, se obține <math> b \in \left\{0,5\right\}</math>. Din <math>3 | \overline{7a8b}</math>, se obține <math> a+b \in \mathcal{M}_3</math>. Pentru <math>b=0</math> rezultă <math> a \in \left\{0,3,6,9\right\}</math>. Pentru <math>b=5</math> rezultă <math> a \in \left\{1,4,7\right\}</math>. Deci<math display="block">A = \left\{ n\in \mathbb{N}, n=\overline{7a8b} , 15 | n \right\} = \left\{ 7080, 7185, 7380, 7485, 7680, 7785, 7980\right\}.</math>Dacă <math>40 | \overline{7x8y}</math>, atunci <math>8 | \overline{7x8y}</math> și <math>5 | \overline{7x8y}</math>. Cum <math>y</math> este par, din <math>5 | \overline{7x8y}</math> se obține <math>y=0</math>. Rezultă <math>\overline{7x80} \, \vdots \, 8</math>, ceea ce implică <math>4x \, \vdots \, 8</math>. Atunci <math> x \in \left\{0,2, 4, 6, 8\right\}</math>. Deci<math display="block">B = \left\{ n\in \mathbb{N}, n=\overline{7x8y} , 40 | n \right\} = \left\{ 7080, 7280, 7480, 7680, 7880\right\}.</math> | ||
Revision as of 04:54, 8 December 2024
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 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} 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{7x8y}} , 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 40} . Să se determine mulțimile 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 \cup B} , 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\cap B} 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 A\setminus B} .
Soluție:
Dacă 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 | \overline{7a8b}} , 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 5 | \overline{7a8b}} 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 3 | \overline{7a8b}} . 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 5 | \overline{7a8b}} , se obține 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 \in \left\{0,5\right\}} . 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 3 | \overline{7a8b}} , se obține 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 \in \mathcal{M}_3} . Pentru 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=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 a \in \left\{0,3,6,9\right\}} . Pentru 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=5} rezultă . Deci
