[tex]2^{n+1}\cdot 5^n + 1 = 2^n\cdot 2\cdot 5^n + 1 = (2\cdot 5)^n\cdot 2 + 1 = 10^n \cdot 2 + 1\\ \\ = \overline{\underbrace{20000\cdots1}_\text{n + 1 cifre}}\text{(daca n = 0 avem doar o singura cifra egala cu 3)}\\ \\ \text{Suma cifrelor} = 2 + 1 = 3 \implies 3 \mid 2^{n+1}\cdot 5^n + 1, \forall n \in \mathbb{N}[/tex]
