E:16888

E:16888 (Gheorghe Boroica)

Considerăm $n$ un număr natural nenul. Demonstrați că numărul $N=\underbrace {44\ldots 4} _{n{\text{ cifre}}}\underbrace {22\ldots 2} _{n{\text{ cifre}}}$ poate fi scris ca produsul a două numere naturale consecutive.

'Soluție

Dacă 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=\underbrace{11\ldots1}_{n \text{ cifre}}} , atunci 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 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).}