I am trying to prove that $\gcd(a, \gcd(b, c)) = \gcd(\gcd(a, b), c)$.
The definition of GCD available to me is as follows:
Given integers a and b, there is one and only one number d with the following properties.
- $d \geqslant 0$
- $d|a$ and $d|b$
- $e|a$ and $e|b$ implies $e|d$.
In the book that I am studying, prime factorization of numbers hasn't been taught yet. Only, the definition of GCD, I've given above has been taught and proven. So, I want to use only this to prove that $\gcd(a, \gcd(b, c)) = \gcd(\gcd(a, b), c)$. Could you please help me?