Let, as usual, Fn and Ln denote the nth Fibonacci number and the nth Lucas number, respectively. In this paper, we consider the Fibonacci numbers F₂, F₃, ..., F_t. Let n ≥ 1 be an integer such that 4n+2 ≤ t ≤ 4n+5 and m = F_2n+2 + F_2n+4 = L_2n+3. We prove that the integers F₂ F_2n+2, F₃F_2n+2, . . . , F_tF_2n+2 modulo m all belong to the interval [F_2n+1, 3F_2n+2]. Furthermore, the endpoints of the interval [F_2n+1, 3F_2n+2] are obtained only by the integers F₄F_2n+2 and F_4n+2F_2n+2, respectively.