Euclidean Algorithm 
Is there always a greatest common divisor of two positive integers?
Every number has at least \(1\) as a divisor.
Hence, every two numbers have at least a common divisor: \(1\).
On the other hand, every number has a finite number of divisors: the divisors of \(n\) are some of the numbers \(1,2,...,n\). So, any two numbers have a finite number of common divisors.
Conclusion: two positive integers always have a finite number of common divisors; there is then one which is bigger than the others.
