E:15777

E:15777 (Anca Mihiș, Baia Mare)

Arătaţi că numărul 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 = \left( 2^2 \cdot 2^4 \cdot 2^6 \cdot \ldots \cdot 2^{2020}\right): \left(2 \cdot 2^3 \cdot 2^5 \cdot \ldots \cdot 2^{2019} \right).} este pătratul unui număr natural.

Soluția 1.

Folosind regulile de calcul cu puteri, 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 A} devine 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^{2-1} \cdot 2^{4-3} \cdot 2^{6-5} \cdot \ldots \cdot 2^{2020-2019}.} 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 A = 2^{1010} = \left( 2^{505} \right)^2,} aşadar 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} este pătratul perfect al numărului 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 2^{505}} .

Soluția 2.

Deîmpărțitul este un 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 1010} factori, 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 2^2 \cdot 2^{4} \cdot 2^{6} \cdot \ldots \cdot 2^{2020} = 2^{\frac{\left(2+2020 \right)\cdot 1010}{2}}=2^{2022\cdot 505}.}

Împărțitorul este un 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 1010} factori, 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 2^1 \cdot 2^{3} \cdot 2^{5} \cdot \ldots \cdot 2^{2019} = 2^{\frac{\left(1+2019 \right)\cdot 1010}{2}}=2^{2020}\cdot 505.} Folosind regulile de calcul cu puteri, numărul $A$ devine

$$A=2^{2022\cdot 505}:2^{2020\cdot 505}.$$

Cum $A=2022\cdot 505-2020\cdot 505=505\cdot \left(2022-2020\right)=505\cdot 2,$ avem că

$$A=2^{505\cdot 2}=\left(2^{505}\right)^{2}.$$

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 A} este pătratul perfect al numărului 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 2^{505}} .