S:E22.136: Difference between revisions
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'' Aflați numerele <math>n</math>, <math>p</math> natuarale neule pentru care <math>2n^2 = 253 \cdot \left(p! + 2022\right)</math>, unde <math>p! = 1\cdot 2\cdot 3\cdot \ldots\cdot p</math>. | '' Aflați numerele <math>n</math>, <math>p</math> natuarale neule pentru care <math>2n^2 = 253 \cdot \left(p! + 2022\right)</math>, unde <math>p! = 1\cdot 2\cdot 3\cdot \ldots\cdot p</math>. | ||
'''Soluție''' | |||
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) = 6</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>. | |||
Deoarece <math>2n^2</math> este număr par, <math>p=1</math> nu convine, deci <math>p \in \left\{ 2,3,4 \right\}</math>. | |||
Cum numerele <math>2</math> și <math>253</math> sunt relativ prime între ele, egalitatea ''<math>2n^2 = 253 \cdot \left(p! + 2022\right)</math>'' are loc doar dacă <math>253 | n</math>, deci există <math>k \in \mathbb{N}^\ast </math> astfel încât <math>n=253 \cdot k </math>. Egalitatea devine <math>2\cdot k^2 = p! + 2022</math>, cu <math>p \in \left\{ 2,3,4\right\}</math>. | |||
Se obține <math>n=506</math> și <math>p = 2</math>. | |||
Latest revision as of 13:22, 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 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 = 253 \cdot \left(p! + 2022\right)} , 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 p! = 1\cdot 2\cdot 3\cdot \ldots\cdot p} .
Soluție
Notăm cu 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) = 6} 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 } .
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 2n^2} este număr par, 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=1} nu convine, deci 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 \in \left\{ 2,3,4 \right\}} .
Cum 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 2} 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 253} sunt relativ prime între ele, egalitatea 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 = 253 \cdot \left(p! + 2022\right)} are loc doar 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 253 | n} , deci există 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 k \in \mathbb{N}^\ast } astfel încât 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=253 \cdot k } . Egalitatea devine 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\cdot k^2 = p! + 2022} , 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 p \in \left\{ 2,3,4\right\}} .
Se obține 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=506} 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 p = 2} .