E:16888

E:16888 (Gheorghe Boroica)

Considerăm 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 n} un număr natural nenul. Demonstrați că 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 N = \underbrace{44\ldots4}_{n \text{ cifre}}\underbrace{22\ldots2}_{n \text{ cifre}} } poate fi scris ca produsul a două numere naturale consecutive.

Soluție

Dacă $a=\underbrace {11\ldots 1} _{n{\text{ cifre}}}$, atunci $9\cdot a+1=10^{n}$ și 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 N= 4\cdot a \cdot 10^n + 2 \cdot a = 2\cdot a \cdot \left(2\cdot 10^n + 1\right) = 2\cdot a \cdot \left(2\cdot \left(9a+1 \right) + 1\right) = 6a \cdot \left(6a+1\right).} În concluzie, 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 N} este produsul numerelor consecutive 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 6a} și 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 6a+1} .