The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_{n + 2} = F_{n + 1} + F_n$. Find the remainder when $F_{1999}$ is divided by 5.

The remainder is 1

Step-by-step explanation:

Given the Fibonacci sequence

F_1 = F_2 = 1, and

F_ (n + 2) = F_ (n + 1) + F_n

We want to find the remainder when F_ (1999) is divided by 5.

Let us write the first 20 numbers of the sequence in (mod 5). They are

F_1 = 1,

F_2 = 1,

F_3 = 2,

F_4 = 3,

F_5 = 5 = 0 (mod 5),

F_6 = 3,

F_7 = 3,

F_8 = 1

F_ (9) = 4

F_ (10) = 0

F_ (11) = 4

F_ (12) = 4

F_ (13) = 3

F_ (14) = 2

F_ (15) = 0

F_ (16) = 2

F_ (17) = 2

F_ (18) = 4

F_ (19) = 1

F_ (20) = 0

We have: 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0

Now, 1999 = 19 (mod 20)

The 19th number in the sequence is 1.

So, the remainder is 1.