2890.11 – Fibonacci Numbers Mod 11

The Fibonacci sequence $\{F_n\}$ is the sequence $1, 1, 2, 3, 4, 8, \ldots$ Determine which members of the sequence are evenly divisible by 11. This requires that you state a rule for the subscripts of such numbers.

Solution

This problem is perfect for solving with a spreadsheet. See the figure below. Notice that the remainders follow a repeating pattern. The Fibonacci numbers that are evenly divisible by 11 occur every 10 terms. They are $F_{10}, F_{20}, F_{30}$ and so forth.