MathNet 2011 Question 75

Let $\left(F_{n}\right)_{n \geq 0}$ be the sequence of Fibonacci numbers: $F_{0}=0$, $F_{1}=1$ and $F_{n+2}=F_{n+1}+F_{n}$, for every $n \geq 0$. Prove that for any prime $p \geq 3$, $p$ divides $F_{2p}-F_{p}$.