E:16892: Difference between revisions

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Created page with "'''E:16892 (Nicolae Mușuroia)''' ''Aflați suma divizorilor pari ai celui mai mare număr natural <math>a</math>, cu <math>a<1000</math>, pentru care suma divizorilor impari este egală cu <math>24</math>.'' '''Soluție''' Căutăm numere de trei cifre, de forma <math>2^m \cdot b</math>, cu <math>m\in \mathbb{N}^\ast</math> și <math>b</math>, unde suma divizorilor numărului natural impar <math>b</math> este egală cu <math>24</math>. Avem două posibilități..."
 
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Căutăm numere de trei cifre, de forma <math>2^m \cdot b</math>, cu <math>m\in \mathbb{N}^\ast</math> și <math>b</math>, unde suma divizorilor numărului natural impar <math>b</math> este egală cu <math>24</math>.
Căutăm numere de trei cifre, de forma <math>2^m \cdot b</math>, cu <math>m\in \mathbb{N}^\ast</math> și <math>b</math>, unde suma divizorilor numărului natural impar <math>b</math> este egală cu <math>24</math>.


Avem două posibilități: <math>24=1+23</math> <math>24=1+3+5+3\cdot 5</math>, deci <math>b=23</math> și <math>b=15</math>.
Avem două posibilități: <math display="block">24=1+23</math>și


Din <math>100 \le 2^m \cdot 23 \le 999</math>, rezultă <math>m\in \left\{3,4,5,6\right\}</math>. Deducem că numerele naturale de trei cifre care îndeplinesc condiția problemei sunt <math>2^3\cdot 23</math>, <math>2^4\cdot 23</math>, <math>2^5\cdot 23</math>, <math>2^3\cdot 15</math>, <math>2^4\cdot 15</math>, <math>2^5\cdot 15</math>, <math>2^6\cdot 15</math>. Dintre acestea, cel mai mare este <math>2^6\cdot 15</math>. Suma divizorilor pari ai numărului <math>a=2^6\cdot 15</math> este <math>S = 2\left(1+3+5+3\cdot 5\right)+2^2\left(1+3+5+3\cdot 5\right)+ \ldots + 2^6 \left(1+3+5+3\cdot 5\right) = 3024.</math>
<math display="block">24=1+3+5+3\cdot 5,</math>deci <math>b=23</math> și <math>b=15</math>.
 
Din <math>100 \le 2^m \cdot 23 \le 999</math>, rezultă <math>m\in \left\{3,4,5,6\right\}</math>.  
 
Deducem că numerele naturale de trei cifre care îndeplinesc condiția problemei sunt  
 
<math>2^3\cdot 23</math>, <math>2^4\cdot 23</math>, <math>2^5\cdot 23</math>, <math>2^3\cdot 15</math>, <math>2^4\cdot 15</math>, <math>2^5\cdot 15</math>, <math>2^6\cdot 15</math>.  
 
Dintre acestea, cel mai mare este <math>2^6\cdot 15</math>. Suma divizorilor pari ai numărului <math>a=2^6\cdot 15</math> este <math display="block">S = 2\left(1+3+5+3\cdot 5\right)+2^2\left(1+3+5+3\cdot 5\right)+ \ldots + 2^6 \left(1+3+5+3\cdot 5\right) = 3024.</math>

Latest revision as of 13:34, 20 September 2025

E:16892 (Nicolae Mușuroia)

Aflați suma divizorilor pari ai celui mai mare număr 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 a} , 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 a<1000} , pentru care suma divizorilor impari este egală 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 24} .

Soluție

Căutăm numere de trei cifre, de forma 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^m \cdot b} , 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 m\in \mathbb{N}^\ast} 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 b} , unde suma divizorilor numărului natural impar 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 b} este egală 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 24} .

Avem două posibilități: 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 24=1+23} 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 24=1+3+5+3\cdot 5,} 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 b=23} 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 b=15} .

Din 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 100 \le 2^m \cdot 23 \le 999} , 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 m\in \left\{3,4,5,6\right\}} .

Deducem că numerele naturale de trei cifre care îndeplinesc condiția problemei 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 2^3\cdot 23} , 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 2^4\cdot 23} , 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 2^5\cdot 23} , 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 2^3\cdot 15} , 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 2^4\cdot 15} , 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 2^5\cdot 15} , 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 2^6\cdot 15} .

Dintre acestea, cel mai mare este 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 2^6\cdot 15} . Suma divizorilor pari ai numărului 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=2^6\cdot 15} este

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