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.