Field Trip! Go watch the Numberphile Video: http://www.numberphile.com/videos/pisano_period.html
The Fibonacci Sequence: 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, …
Many mathematicians have studied this
integer sequence for hundreds of years and found several interesting
patterns. One of them is a pattern
called Pisano periods.
It says that the m^{th} Fibonacci
number (or F(m)) evenly divides the n^{th} Fibonacci number (or F(n)) if
m evenly divides n. So the 3^{rd}
Fibonacci number (or F(3) which equals 2) divides every 3^{rd}
Fibonacci number ((F(6), F(9), F(12), etc.)
1 1
2 1
3 2
4 3
5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 
2 is the 3^{rd} Fibonacci
number. Every 3^{rd} number
after 2 is divisible by 2 (it’s and even number).
3 is the 4^{th} Fibonacci
number. Every 4^{th} number
after 3 is divisible by 3.
5 is the 5^{th} Fibonacci
number. Every 5^{th} number
after 5 is divisible by 5.
8 is the 6^{th} Fibonacci
number. Every 6^{th} number
after 8 is divisible by 8.
13 is the 7^{th} Fibonacci
number. Every 7^{th} number
after 13 is divisible by 13.
21 is the 8^{th} Fibonacci
number. Every 8^{th} number
after 21 is divisible by 21.
34 is the 9^{th} Fibonacci number. Every 9^{th} number after 34 is
divisible by 34.
55 is the 10^{th} Fibonacci
number. Every 10^{th} number
after 55 is divisible by 55.
89 is the 11^{th} Fibonacci
number. Every 11^{th} number
after 89 is divisible by 89.
144 is the 12^{th} Fibonacci
number. Every 12^{th} number
after 144 is divisible by 144.
Etc.

David
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