E:16892

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 $b$, unde suma divizorilor numărului natural impar $b$ este egală cu $24$.

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

Din $100\leq 2^{m}\cdot 23\leq 999$, rezultă $m\in \left\{3,4,5,6\right\}$. Deducem că numerele naturale de trei cifre care îndeplinesc condiția problemei sunt $2^{3}\cdot 23$, $2^{4}\cdot 23$, $2^{5}\cdot 23$, $2^{3}\cdot 15$, 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 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 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.}