E:15345: Difference between revisions
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'''Soluție''' | '''Soluție''' | ||
Ecuația se scrie în mod echivalent | Ecuația se scrie în mod echivalent<math display="block"> x^y \cdot x^z + x^y + x^z = 584,</math>ceea ce conduce la <math display="block">x^y \cdot x^z + x^y + x^z= 585.</math>De aici avem <math>x^y \cdot(x^z+1) + (x^z + 1) = 585</math>, de unde rezultă <math display="block">(x^y + 1) \cdot (x^z + 1) = 585.</math>Deoarece <math>585</math> este număr impar deducem că cele două paranteze sunt numere impare; mai mult <math>x</math> este număr par. | ||
<math> x^y | Cum <math> 585 = 3 \cdot 195 = 5 \cdot 117 = 9 \cdot 65 = 13 \cdot 45 = 15 \cdot 39</math> putem avea <math>x^y + 1 = 3</math> și <math> x^z + 1 = 195; x^y + 1 = 5 </math> și <math> x^z + 1 = 117; x^y + 1 = 9 </math> și <math> x^z + 1 = 65; x^y + 1 = 13 </math> și <math> x^z + 1 = 45; x^y + 1 = 15 </math> și <math> x^z + 1 = 39 </math>, sau invers. | ||
Soluții naturale obținem numai pentru <math> x^y + 1 = 9 </math> sau <math> x^y + 1 = 65 </math>. | |||
<math>x^y | |||
sau <math> | |||
Găsim <math> x = 2, y = 3, z = 6 </math>, sau <math> x = 2, y = 6, z = 3 </math> | Găsim <math> x = 2, y = 3, z = 6 </math>, sau <math> x = 2, y = 6, z = 3 </math> | ||
Revision as of 15:35, 20 August 2025
E:15345 (Călin Dănuț Hossu, Baia Mare)
Determinați numerele 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{xyz}} , scrise în baza 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 10} , știind 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 x^{y+z} + x^y + x^z - 584 = 0} .
Soluție
Ecuația se scrie în mod echivalentFailed 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 \cdot x^z + x^y + x^z = 584,} ceea ce conduce la 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 \cdot x^z + x^y + x^z= 585.} De aici avem , de unde rezultă
Cum 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 585 = 3 \cdot 195 = 5 \cdot 117 = 9 \cdot 65 = 13 \cdot 45 = 15 \cdot 39} putem avea 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 = 3} 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 x^z + 1 = 195; x^y + 1 = 5 } 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 x^z + 1 = 117; x^y + 1 = 9 } 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 x^z + 1 = 65; x^y + 1 = 13 } 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 x^z + 1 = 45; x^y + 1 = 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 x^z + 1 = 39 } , sau invers.
Soluții naturale obținem numai 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 x^y + 1 = 9 } sau 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 = 65 } .
Găsim 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 = 2, y = 3, z = 6 } , sau 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 = 2, y = 6, z = 3 }