This means that the \(\gcd\left(89,144\right)=1.\) Notice from the Euclidean Algorithm above that every remainder we have for each equation represents a Fibonacci number and going from bottom to top, we have generated the Fibonacci Sequence. This will occur when we are trying to find the greatest common divisor of two consecutive Fibonacci numbers. With that in mind, we can also repeat this process for nonconsecutive Fibonacci numbers. In either case, the last nonzero remainder produced will always result in a solution that is also a Fibonacci number.
- Another unique application with the use of the Fibonacci sequence is the conversion between miles to kilometers. This a unique application that can be applied in real world applications and the math behind it is very simple. We can take any Fibonacci number and multiply it by the conversion factor for miles to kilometers, which is \(1.609.\) The result will produce a decimal , which when rounded up to the next nearest whole number will be the next consecutive Fibonacci number in the sequence. For example, if we wanted to see how many kilometers is in \(13\) miles we can compute \(13\cdot1.609=20.917\). When rounding this to the next nearest whole number, we get that \(21\) kilometers is in \(13\ \) miles, which so happens to be the next Fibonacci number in our sequence.
- The reason this mathematical application works is due to the fact that Fibonacci numbers have a unique property where the ratio between any two consecutive Fibonacci numbers tends to the Golden ratio, \(1.618\) as the numbers increase. Since there are \(1.609\) kilometers in a mile, we can see this conversion factor tends to the Golden ratio.
To generalize conversion, we can take any two Fibonacci numbers, \(F_{n+1}\) and \(F_n\). Their ratio is: