P:1795

P:1795 (Gheorghe Boroica)

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 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 $2025$ de factori:

$$42=2\cdot 3\cdot 7\cdot {\stackrel {1\cdot 1\cdot \ldots \cdot 1}{2022}}$$

$$42=6\cdot 7\cdot {\stackrel {1\cdot 1\cdot \ldots \cdot 1}{2023}}$$

$$42=2\cdot 21\cdot {\stackrel {1\cdot 1\cdot \ldots \cdot 1}{2023}}$$

$$42=3\cdot 14\cdot {\stackrel {1\cdot 1\cdot \ldots \cdot 1}{2023}}$$

$$42=42\cdot {\stackrel {1\cdot 1\cdot \ldots \cdot 1}{2024}}$$

Pentru fiecare dintre aceste produse se obțin următoarele sume ale factorilor lor:

$$S_{1}=2+3+7+{\stackrel {1+1+\ldots +1}{2022}}=12+2022=2034,$$

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

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

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

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

În concluzie, suma minimă 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 2034} se obține pentru produsul 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 \stackrel{1\cdot 1 \cdot \ldots \cdot 1}{2022}} .