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First question can be solved by Induction
you may view the answer on this PDF:
http://www.math.ucdavis.edu/~nakamur...math55fibo.pdf

Lemma 5

the other part can be solved by contradiction:
Assume that there exist some two consecutive Fibonacci numbers say

fn and fn+1 that have a common divisor d, where d is greater than 1.

Thus, their difference fn+1 - fn = fn-1 will also be divisible by d.

However, we know that f1=1 which is clearly not divisible by d. Thus, we have reached a contradiction. Therefore, consecutive Fibonacci numbers are relatively prime.

Second question was proved by Euler:
Euler found that the fifth Fermat number, 4294967297, is divisible by 641. Therefore, Fermat's assertion is false.
F5=4294967297=641*6700417
You may refer to the following page:
http://mathworld.wolfram.com/FermatNumber.html