Answer:
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.
