S:E22.136: Difference between revisions

Revision as of 13:09, 5 December 2024

S:E22.136 (Cristina Vijdeluc, Mihai Vijdeluc)

Aflați 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 n} , 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 p} natuarale neule pentru care $2n^{2}=253\cdot \left(p!+2022\right)$ , unde $p!=1\cdot 2\cdot 3\cdot \ldots \cdot p$ .

Soluție

Notăm 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 u\left( N \right)} ulitima cifră a numărului natural 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 N} .

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 p \ge 5 } , 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 u\left( 253 \cdot \left(p! + 2022\right) \right) = 625} 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 u\left( 2n^2 \right) \in \left\{ 0,2,8 \right\}} . Deci, 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 p \ge 5 } , 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 2n^2 \ne 253 \cdot \left(p! + 2022\right)} oricare ar fi 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 n \in \mathbb{N}^\ast } .

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 '''Soluție'''
-Notăm cu <math>u\left( N \right)</math> ulitima cifră a numărului natural
+Notăm cu <math>u\left( N \right)</math> ulitima cifră a numărului natural <math>N</math>.
+Pentru <math> p \ge 5 </math>, avem <math>u\left( 253 \cdot \left(p! + 2022\right) \right) = 625</math> și <math> u\left( 2n^2 \right) \in \left\{ 0,2,8 \right\}</math>. Deci, pentru <math> p \ge 5 </math>, avem  <math>2n^2 \ne 253 \cdot \left(p! + 2022\right)</math> oricare ar fi <math>n \in \mathbb{N}^\ast </math>.