E:15695: Difference between revisions

Latest revision as of 12:57, 30 December 2023

E:15695 (Cristina Vijdeluc şi Mihai Vijdeluc)

Aflaţi numerele 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{ab},} ştiind 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 \overline{aaa} \cdot b + \overline{bb} = 2020.}

Soluție:

Relaţia dată se scrie 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 111 \cdot a \cdot b + 11 \cdot b = 2020} și în mod echivalent 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 \cdot (111a + 11) = 2020.} Divizorii lui 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 2020} sunt 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 1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010} şi $2020$ .

Cum $111a+11\geq 122,$ sunt posibile cazurile:

$111a+11=202$ şi $b=10$

$111a+11=404$ şi $b=5$

$111a+11=505$ şi $b=4$

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 111a + 11 = 1010} şi 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 = 2}

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 111a + 11 = 2020} şi 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 = 1.}

Convine numai cazul 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 111a + 11 = 1010} şi 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 = 2,} 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 = 9} şi 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 = 2.} 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 \overline{ab} = 92 }

@@ Line 1: / Line 1: @@
-'''E:15695 (Cristina Vijdeluc şi Mihai Vijdeluc, Baia Mare)'''
+'''E:15695 (Cristina Vijdeluc şi Mihai Vijdeluc)'''
 ''Aflaţi numerele de forma ''<math>\overline{ab},</math>'' ştiind că ''<math>\overline{aaa} \cdot b + \overline{bb} = 2020.</math>
@@ Line 5: / Line 5: @@
 '''Soluție:'''
-Relaţia dată se scrie <math>111 \cdot a \cdot b + 11 \cdot b = 2020</math> sau <math>b(111a + 11) = 2020</math>. Divizorii lui <math>2020</math> sunt <math>1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010</math> şi <math>2020</math>. Cum <math>111a + 11 \ge 122,</math> sunt posibile cazurile:<math>111a + 11 = 202</math> şi <math>b = 10, 111a + 11 = 404</math> şi <math>b = 5, 111a + 11 = 505</math> şi <math>b = 4, 111a + 11 = 1010</math> şi <math>b = 2</math> sau <math>111a + 11 = 2020</math> şi <math>b = 1.</math> Convine numai cazul <math>111a + 11 = 1010</math> şi <math>b = 2,</math> de unde <math>a = 9</math> şi <math>b = 2.</math>
+Relaţia dată se scrie <math>111 \cdot a \cdot b + 11 \cdot b = 2020</math> și în mod echivalent <math display="block">b \cdot (111a + 11) = 2020.</math>Divizorii lui <math>2020</math> sunt <math>1, 2, 4, 5, 10, 20, 101, 202, 404, 505, 1010</math> şi <math>2020</math>.
+Cum <math>111a + 11 \ge 122,</math> sunt posibile cazurile:
+<math>111a + 11 = 202</math> şi <math>b = 10</math>
+<math>111a + 11 = 404</math> şi <math>b = 5</math>
+<math>111a + 11 = 505</math> şi <math>b = 4</math>
+<math>111a + 11 = 1010</math> şi <math>b = 2</math>
+<math>111a + 11 = 2020</math> şi <math>b = 1.</math>
+Convine numai cazul <math>111a + 11 = 1010</math> şi <math>b = 2,</math> de unde <math>a = 9</math> şi <math>b = 2.</math> Deci<math display="block">\overline{ab} = 92 </math>