ZA-SAMO 1999 Problem 4

The sequence $L_1,\ L_2,\ L_3,\ \dots$ is defined by $$L_1 = 1,\ \ L_2 = 3,\ \ L_n = L_{n - 1} + L_{n - 2}\textrm{ for }n > 2.$$ Prove that $L_p - 1$ is divisible by $p$ if $p$ is prime.