KöMaL 2012 C1137

The first two terms of the Fibonacci sequence are $a_1$=1, $a_2$=1, and every further term equals the sum of the two preceding terms, that is, $a_n$=$a_{n-2}$+$a_{n-1}$ ($n \ge$3). Prove that the sequence has no term that leaves a remainder of 4 when divided by 13.