S:E18.122: Difference between revisions
Created page with "'''S:E18.122 (Florin Bojor)''' ''Verificați dacă numărul <math>a={1^{{{3}^{{{5}^{{{\cdots}^{2019}}}}}}}}+{3^{{{5}^{{{7}^{{{\cdots}^{2019}}}}}}}}+{5^{{{7}^{{{9}^{{{\cdots}^{2019}}}}}}}}+\ldots +{2017^{2019}}+2019</math> este divizibil cu <math>10</math>. '''Soluție.''' Notăm cu <math>u\left(n\right)</math> ultima cifră a numărului natural <math>n</math>, iar prin <math>\tilde{n}</math> nătăm numărul" |
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'''Soluție.''' | '''Soluție.''' | ||
Notăm cu <math>u\left(n\right)</math> ultima cifră a numărului natural <math>n</math>, iar prin <math>\ | Notăm cu <math>u\left(n\right)</math> ultima cifră a numărului natural <math>n</math>, iar prin <math>\overline{n}</math> notăm numărul <math>n^{{n+2}^{{n+4}^{\ldots 2019}}}</math>. Cu această notație, avem<math display="block">a=\overline{1}+\overline{3}+\overline{5}+\ldots+\overline{2019}</math>este o sumă cu 1010 tremeni. | ||
Dacă <math>u\left(n\right) = 1</math>, atunci <math>u\left(n^x\right) = 1</math> oricare ar fi <math>x\in \mathbb{N}</math>. Deci<math display="block">u\left(\overline{1}\right) = u\left(\overline{11}\right) = u\left(\overline{21}\right) = \ldots = u\left(\overline{2011}\right) = 1,</math>sunt <math>202</math> astfel de numere. | |||
Dacă <math>u\left(n\right) = 5</math>, atunci <math>u\left(n^x\right) = 5</math> oricare ar fi <math>x\in \mathbb{N}</math>. Deci<math display="block">u\left(\overline{5}\right) = u\left(\overline{15}\right) = u\left(\overline{25}\right) = \ldots = u\left(\overline{2015}\right) = 5,</math>sunt <math>202</math> astfel de numere. | |||
Cum <math>u\left(9^{2x+1}\right) = 9</math> oricare ar fi <math>x\in \mathbb{N}</math>, avem că<math display="block">u\left(\overline{9}\right) = u\left(\overline{19}\right) = u\left(\overline{29}\right) = \ldots = u\left(\overline{2019}\right) = 9,</math>sunt <math>202</math> astfel de numere. | |||
Cum <math>u\left(3^{4x+1}\right)=3</math> și <math>u\left(3^{4x+3}\right)=7</math> oricare ar fi <math>x\in \mathbb{N}</math>, avem <math>101</math> pentru care au loc egalitățile<math display="block">u\left(\overline{3}\right) = u\left(\overline{23}\right) = u\left(\overline{43}\right) = \ldots = u\left(\overline{2003}\right) = 3</math>și <math>101</math> pentru care au loc egalitățile<math display="block">u\left(\overline{13}\right) = u\left(\overline{33}\right) = u\left(\overline{53}\right) = \ldots = u\left(\overline{2013}\right) = 7.</math>Cum <math>u\left(7^{4x+1}\right)=7</math> și <math>u\left(7^{4x+3}\right)=3</math> oricare ar fi <math>x\in \mathbb{N}</math>, avem <math>101</math> pentru care au loc egalitățile<math display="block">u\left(\overline{7}\right) = u\left(\overline{27}\right) = u\left(\overline{47}\right) = \ldots = u\left(\overline{2007}\right) = 7</math>și <math>101</math> pentru care au loc egalitățile<math display="block">u\left(\overline{17}\right) = u\left(\overline{37}\right) = u\left(\overline{57}\right) = \ldots = u\left(\overline{2017}\right) = 3.</math>În concluzie, se obține<math display="block">u\left(a\right) = u\left( 202\cdot 1 + 101 \cdot \left(3+7\right) + 202\cdot 5 + 101\cdot \left(7+3\right)+202\cdot 9\right) = 0,</math>ceea ce implcă faptul că numărul <math>a</math> este divizibil cu <math>10</math>. | |||
Latest revision as of 19:15, 14 July 2025
S:E18.122 (Florin Bojor)
Verificați dacă numărul 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={1^{{{3}^{{{5}^{{{\cdots}^{2019}}}}}}}}+{3^{{{5}^{{{7}^{{{\cdots}^{2019}}}}}}}}+{5^{{{7}^{{{9}^{{{\cdots}^{2019}}}}}}}}+\ldots +{2017^{2019}}+2019} este 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 10} .
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)} ultima 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} , iar prin 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{n}} notăm numărul 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^{{n+2}^{{n+4}^{\ldots 2019}}}} . Cu această notație, avemFailed 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=\overline{1}+\overline{3}+\overline{5}+\ldots+\overline{2019}} este o sumă cu 1010 tremeni.
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 u\left(n\right) = 1} , 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 u\left(n^x\right) = 1} 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 x\in \mathbb{N}} . DeciFailed 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(\overline{1}\right) = u\left(\overline{11}\right) = u\left(\overline{21}\right) = \ldots = u\left(\overline{2011}\right) = 1,} 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 202} astfel de numere.
Dacă , atunci oricare ar fi . Deci
Cum oricare ar fi , avem că
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 u\left(3^{4x+1}\right)=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 u\left(3^{4x+3}\right)=7} 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 x\in \mathbb{N}} , 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 101} pentru care au loc egalitățileFailed 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(\overline{3}\right) = u\left(\overline{23}\right) = u\left(\overline{43}\right) = \ldots = u\left(\overline{2003}\right) = 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 101} pentru care au loc egalitățileFailed 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(\overline{13}\right) = u\left(\overline{33}\right) = u\left(\overline{53}\right) = \ldots = u\left(\overline{2013}\right) = 7.} 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 u\left(7^{4x+1}\right)=7} 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(7^{4x+3}\right)=3} 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 x\in \mathbb{N}} , 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 101} pentru care au loc egalitățileFailed 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(\overline{7}\right) = u\left(\overline{27}\right) = u\left(\overline{47}\right) = \ldots = u\left(\overline{2007}\right) = 7} 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 101} pentru care au loc egalitățileFailed 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(\overline{17}\right) = u\left(\overline{37}\right) = u\left(\overline{57}\right) = \ldots = u\left(\overline{2017}\right) = 3.} În concluzie, se obțineFailed 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(a\right) = u\left( 202\cdot 1 + 101 \cdot \left(3+7\right) + 202\cdot 5 + 101\cdot \left(7+3\right)+202\cdot 9\right) = 0,} ceea ce implcă faptul că numărul 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} este 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 10} .