For a division expression where $a$ and $b$ are two positive integers such that $a>b$ we can write it as :
$a = bq + r$ ($q$ is quotient and $r$ is remainder from division)
While finding GCD of $a$ and $b$, I come to an intuitive conclusion that the common factor $k$ of $a$ and $b$ should satisfy : $\frac{1}{2}b> k$ and $\frac{1}{2}r>$ $k$ if $r$ is a positive non-zero integer.
I want to prove to myself that $k$ must be smaller than $r$ and that there exists no common factor if $r$ is non-zero positive integer such that $k > r$.
To do so, I used the basic principle :
$$\text{gcd}(a,b) = \text{gcd}(b,r)$$
Therefore, the common factor of a and b must be smaller than or equal to $r$ because you cannot have a factor greater than number itself.
I want to know if there is any other possibility of arriving at a contradiction if we start with a premise that $k > r$ when $r>0$ .