Let $a$ and $b$ be integers, not both zero. Prove that for a positive integer $d$, $d=gcd(a, b)$ If and only if
1) $d$ divides $a$ ($d\vert a$) and $d$ divides $b$ ($d\vert b$) 2) whenever $c$ divides $a$ and $c$ divides $b$, then $c$ divides $d$ ..
How to deduce this theorem? I'm new to this stuff. Can somebody give a proper direction within this mathematics field?
That depends on your definition of $\gcd(a,b)$. If you define it to be generator of $a\mathbb Z+b\mathbb Z$, then all of them (even Bézout's identity) follows directly from definition.
Also, that is clear from the name of $\gcd(a,b)$, the first statement says that $\gcd(a,b)$ is a common factor of $a$ and $b$, and the last statement says that it is the greatest.
We have $d=GCD(a,b)\implies a=dy\land b=dz$, where $GCD(y,z)=1$. We also have $c|(a,b)\implies a=c(ky)$ and $b=c(kz)$ where $k\in\mathbb{N}$. This means $d=ck$ thus c|d.
