P:1795

P:1795 (Gheorghe Boroica)

Numărul $42$ se scrie ca și produsul a 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 2025} numere naturale. Determinați suma minimă a tuturor factorilor acestui produs.

Soluție

Sunt posibile următoarele scrieri ale 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 42} ca și produs de 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 2025} de factori: 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 42 = 2 \cdot 3 \cdot 7 \cdot \underbrace{1\cdot 1 \cdot \ldots \cdot 1}_{2022}} 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 42 = 6 \cdot 7 \cdot \underbrace{1\cdot 1 \cdot \ldots \cdot 1}_{2023}} 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 42 = 2 \cdot 21 \cdot \underbrace{1\cdot 1 \cdot \ldots \cdot 1}_{2023}} 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 42 = 3 \cdot 14 \cdot \underbrace{1\cdot 1 \cdot \ldots \cdot 1}_{2023}} 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 42 = 42 \cdot \underbrace{1\cdot 1 \cdot \ldots \cdot 1}_{2024}} Pentru fiecare dintre aceste produse se obțin următoarele sume ale factorilor lor: 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_1 = 2 + 3 + 7 + \underbrace{1+ 1 + \ldots + 1}_{2022} = 12+2022 = 2034,}

$$S_{2}=6+7+\underbrace {1+1+\ldots +1} _{2023}=13+2023=2036,$$

$$S_{3}=2+21+\underbrace {1+1+\ldots +1} _{2023}=23+2023=2046,$$

$$S_{4}=3+14+\underbrace {1+1+\ldots +1} _{2023}=17+2023=2040,$$

$$S_{5}=42+\underbrace {1+1+\ldots +1} _{2024}=42+2024=2066.$$

În concluzie, suma minimă $2034$ se obține pentru produsul $42=2\cdot 3\cdot 7\cdot \underbrace {1\cdot 1\cdot \ldots \cdot 1} _{2022}$.