Playing With The Numbers

One of the more interesting things about the Fibonacci sequence is the interplay between the numbers; how they interrelate and their "behaviors." Some of the interactions are simple fun tricks, such as those in the children's book where I originally discovered the sequence to more complex paradoxes.

A few examples:

Divisible by 11

The sum of any ten consecutive Fibonacci numbers is always evenly divisible by 11.

1	5	89
1	8	144
2	13	233
3	21	377
5	34	610
8	55	987
13	89	1,597
21	144	2,584
34	233	4,1841
+55	+377	+6,765
_____	_____	_______
143 / 11=13	979 / 11=89	17,567 / 11=1,579

More divisibility

Here's another trick. As the consecutive integers (Fn) increase, note how they are divisible by consecutive Fibonacci numbers:

Every 3rd Fibonacci number is divisible by 2.
Every 4th Fibonacci number is divisible by 3.
Every 5th Fibonacci number is divisible by 5.
Every 6th Fibonacci number is divisible by 8.
Every 7th Fibonacci number is divisible by 13.
Every 8th Fibonacci number is divisible by 21.

Factors of Fibonacci

Another interesting characteristic of the Fibonacci sequence is that no two consecutive Fibonacci numbers have any common factors. Like this:

Fibonacci number and Prime Factors
Fibonacci Number	Prime Factors	Fibonacci Number	Prime Factors
1	1	55	5x11
1	1	89	89
2	2	144*	2x2x2x2x3x3*
3	3	233	233
5	5	377	13x29
8	2x2x2	610	2x5x61
13	13	987	3x7x47
21	3x7	1597	1579
34	2x17	2584	2x2x2x17x19

* The 12th Fibonacci number (144) is the square of 12.
It is also the only square number in the entire sequence
well, as long as you don't count the number 1.