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MathPassion wrote at 2016-02-08 00:36:18
Just do MOD 8 of the given Fibonacci sequence starting with 1.

1 MOD 8 congruent to 1, 8 MOD 8 is congruent to 0 and so on.



    1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 1, 0, 1, 1, 2, 3, 5, 0...

         (Repetition after 12 terms)

There is cyclicity of 12.

Since 2009 MOD 12 congruent to 5  (Remainder when 2009 is divided by 12).



The 5th term in the MOD 8 sequence (1,1,2,3,5,0,5,5,2...) is 5.



Therefore remainder when 2009th terms is divided by 8 is 5.


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