E:16380: Difference between revisions
Pagină nouă: '''E:16380 (Cristina Vijdeluc, Salonic şi Mihai Vijdeluc, Baia Mare)''' ''Aflaţi numerele naturale ''<math>a,b,c,d</math>'' pentru care are loc relaţia ''<math>2(3^{a + 1} + 3^{b + 1} + 3^{c + 1}) = 3 \cdot 6 \cdot 9 \cdot \ldots \cdot d.</math> '''Soluție:''' Egalitatea din enunţul se poate scrie <math>6(3^a + 3^b +3^c) = 3 \cdot 6 \cdot 9 \cdot \ldots \cdot d.</math> Trebuie ca <math>d \geqslant 6</math> şi <math>d</math> să fie divizibil cu <math>3.</math> Dacă... |
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Dacă <math>d = 12,</math> obţinem <math>3^a + 3^b +3^c = 3^4 \cdot 4,</math> adică <math>3^{a - 4} + 3^{b - 4} + 3^{c - 4} = 4,</math> ecuaţia care nu are soluţii. | Dacă <math>d = 12,</math> obţinem <math>3^a + 3^b +3^c = 3^4 \cdot 4,</math> adică <math>3^{a - 4} + 3^{b - 4} + 3^{c - 4} = 4,</math> ecuaţia care nu are soluţii. | ||
Pentru <math>d \geqslant 15,</math> ultima cifră a produsului din membrul drept este zero. Dar <math>u(3^n) \in {1,3,7,9},</math> deci o sumă de trei puteri ale lui <math>3</math> nu are niciodată ultima cifră zero. | Pentru <math>d \geqslant 15,</math> ultima cifră a produsului din membrul drept este zero. Dar <math>u(3^n) \in \{1,3,7,9\},</math> deci o sumă de trei puteri ale lui <math>3</math> nu are niciodată ultima cifră zero. | ||
Aşadar, soluţiile căutate sunt <math>(a, b, c, d) \in \{(0, 0, 0, 6),(2, 2, 2, 9)\}.</math> | Aşadar, soluţiile căutate sunt <math>(a, b, c, d) \in \{(0, 0, 0, 6),(2, 2, 2, 9)\}.</math> | ||
Revision as of 20:17, 27 December 2023
E:16380 (Cristina Vijdeluc, Salonic şi Mihai Vijdeluc, Baia Mare)
Aflaţi numerele naturale 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,c,d} pentru care are loc relaţia 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 2(3^{a + 1} + 3^{b + 1} + 3^{c + 1}) = 3 \cdot 6 \cdot 9 \cdot \ldots \cdot d.}
Soluție:
Egalitatea din enunţul se poate scrie Trebuie ca 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 d \geqslant 6} 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 d} să fie divizibil 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 3.}
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 d = 6,} 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 3^a + 3^b +3^c = 3,} ceea ce înseamnă 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 = c = 0.}
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 d = 9,} 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 3^a + 3^b +3^c = 27,} de unde rezultă 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 = b = c = 2.}
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 d = 12,} obţinem 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^a + 3^b +3^c = 3^4 \cdot 4,} adică 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^{a - 4} + 3^{b - 4} + 3^{c - 4} = 4,} ecuaţia care nu are soluţii.
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 d \geqslant 15,} ultima cifră a produsului din membrul drept este zero. Dar 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 u(3^n) \in \{1,3,7,9\},} deci o sumă de trei puteri ale 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 3} nu are niciodată ultima cifră zero.
Aşadar, soluţiile căutate 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 (a, b, c, d) \in \{(0, 0, 0, 6),(2, 2, 2, 9)\}.}