Prove that for every integer $n \geq 0$, the number $4^{2n+1}+3^{n+2}$ is a multiple of 13.

Question

Prove that for every integer $n \geq 0$, the number $4^{2n+1}+3^{n+2}$ is a multiple of 13.

Answer 1

HINT: we have $$4^{2n+1}+3^{n+2}=16^n\cdot 4+3^n\cdot 9\equiv 3^n\cdot 4+3^n\cdot 9\equiv 3^n\cdot 13$$