To find $F_{2016} \mod 5$, where $F_n$ is the Fibonacci sequence, we can use the periodicity property of the Fibonacci sequence modulo any integer. This property is called the Pisano period.

Step 1: Determine the Pisano Period for Modulo 5

The Pisano period $\pi(5)$ is the length of the repeating sequence of the Fibonacci numbers modulo 5. Let's calculate this period:

Start with the Fibonacci sequence modulo 5:

$F_0 = 0, \quad F_1 = 1$ $F_2 = (F_1 + F_0) \mod 5 = (1 + 0) \mod 5 = 1$ $F_3 = (F_2 + F_1) \mod 5 = (1 + 1) \mod 5 = 2$ $F_4 = (F_3 + F_2) \mod 5 = (2 + 1) \mod 5 = 3$ $F_5 = (F_4 + F_3) \mod 5 = (3 + 2) \mod 5 = 0$ $F_6 = (F_5 + F_4) \mod 5 = (0 + 3) \mod 5 = 3$ $F_7 = (F_6 + F_5) \mod 5 = (3 + 0) \mod 5 = 3$ $F_8 = (F_7 + F_6) \mod 5 = (3 + 3) \mod 5 = 1$ $F_9 = (F_8 + F_7) \mod 5 = (1 + 3) \mod 5 = 4$ $F_{10} = (F_9 + F_8) \mod 5 = (4 + 1) \mod 5 = 0$ $F_{11} = (F_{10} + F_9) \mod 5 = (0 + 4) \mod 5 = 4$ $F_{12} = (F_{11} + F_{10}) \mod 5 = (4 + 0) \mod 5 = 4$ $F_{13} = (F_{12} + F_{11}) \mod 5 = (4 + 4) \mod 5 = 3$ $F_{14} = (F_{13} + F_{12}) \mod 5 = (3 + 4) \mod 5 = 2$ $F_{15} = (F_{14} + F_{13}) \mod 5 = (2 + 3) \mod 5 = 0$

However, we can notice the sequence starts repeating from $F_15$, and more generally:

The Pisano period for 5 is $\pi(5) = 20$.

Step 2: Find $F_{2016} \mod 5$

Since the Fibonacci sequence modulo 5 repeats every 20 terms, we can reduce 2016 modulo 20: $2016 \mod 20 = 16$ So, $F_{2016} \mod 5 = F_{16} \mod 5$.

Step 3: Determine $F_{16} \mod 5$

From our earlier calculation: $F_{16} = (F_{15} + F_{14}) \mod 5 = (0 + 2) \mod 5 = 2$

Therefore, $F_{2016} \mod 5 = 2$.

Final Answer:

$F_{2016} \mod 5 = 2$

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the Pisano period for other moduli, such as 7 or 10?
2. How can we use Pisano periods to compute large Fibonacci numbers modulo any integer?
3. How does the Pisano period change as the modulus changes?
4. Can you derive the closed form expression for Fibonacci numbers using matrix exponentiation?
5. What applications do Pisano periods have in cryptography?

Tip: The Pisano period is a powerful tool to simplify calculations involving Fibonacci numbers mod $n$, especially for large $n$.