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 $2025$ numere naturale. Determinați suma minimă a tuturor factorilor acestui produs.

Soluție

Sunt posibile următoarele scrieri ale numărului $42$ 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:

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

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

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_3 = 2 + 21 + \stackrel{1+ 1 + \ldots + 1}{2023} = 23+2023 = 2046,} 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_4 = 3 + 14 + \stackrel{1+ 1 + \ldots + 1}{2023} = 17+2023 = 2040,} 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_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}} .