E:15345: Difference between revisions

Latest revision as of 15:37, 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 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+1) + (x^z + 1) = 585} , de unde 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 (x^y + 1) \cdot (x^z + 1) = 585.} 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 585} este număr impar deducem că cele două paranteze sunt numere impare; mai mult 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} este număr par.

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 } .

În concluzie, am obținut 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} = 236} , respectiv 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} = 263} .

@@ Line 5: / Line 5: @@
 '''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 \cdot x^z + x^y + x^z = 584</math>,
+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.
-sau <math>x^y \cdot x^z + x^y + x^z= 585.</math>
-De aici
+Soluții naturale obținem numai pentru <math> x^y + 1 = 9 </math> sau <math> x^y + 1 = 65 </math>.
-<math>x^y \cdot(x^z+1) + (x^z + 1) = 585</math>
-sau <math>(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. 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>.
+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>
+În concluzie, am obținut <math>\overline{xyz} = 236</math>, respectiv <math>\overline{xyz} = 263</math>.