Fibonacci QuestA number of self marking quizzes based on the fascinating Fibonacci Sequence. 
This is level 5; Investigate the highest common factor of every nth term of the Fibonacci sequence..
With the exceptions of 1, 8 and 144, every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number. This is known as Carmichael's theorem.
Here's a quick trick for roughly converting between miles and kilometres using the Fibonacci sequence.
To convert from miles to kilometers find the term of the sequence closest to the distance in miles. The next term in the sequence is an approximation of that distance in kilometres.
To convert from kilometres to miles find the term of the sequence closest to the distance in kilometres. The previous term in the sequence is an approximation of that distance in miles.
Can you explain why this trick works?
There is an online exercise providing practice converting between miles and kilometres here.
This is Fibonacci Quest level 5. You can also try:
Level 1
Level 2
Level 3
Level 4
Level 6
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Level 1  Continue the basic Fibonacci sequence
Level 2  Continue the Fibonacci sequence in reverse
Level 3  Find algebraic expressions for each term of a Fibonacci sequence
Level 4  Finding the ratio of two successive numbers in Fibonacci's sequence
Level 5  Investigate the highest common factor of every nth term of the Fibonacci sequence.
Level 6  Finding missing terms from Fibonaccitype sequences.