Question

19 Arătați că (2^n+2 + 2n) | (4^n+1 +4^n), pentru orice număr natural n. Dau coroana!!!:)​

Answer 1

$\it 2^{n+2}+2^n=2^n\cdot2^2+2^n=2^n(4+1)=2^n\cdot5\\ \\ 4^{n+1}+4^n=4^n\cdot4+4^n=4^n(4+1)=4^n\cdot5\\ \\ 2\Big|4 \Rightarrow 2^n\Big|4^n \Rightarrow 2^n\cdot5\Big|4^n\cdot5 \Rightarrow (2^{n+2}+2^n)\Big|(4^{n+1}+4^n), \forall\ n\in\mathbb{N}$