Abstract In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L2mn+k ≡ (-1)(m+1)nLk (mod Lm), F2mn+k ≡ (-1)(m+1)nFk (mod Lm), L2mn+k ≡ (-1)mnL2mn+k(mod Fm) and F2mn+k ≡ (-1)mnFk (mod Fm). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number Ln such that Ln = L2ktLmx2 for m > 1 and k ≥ 1. Moreover it is proved that Ln = LmLr is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.