I was referring Elementary Number Theory and its Applications by Kenneth Rosen (6th Ed., p. $95$).
My main object is to understand a proof of the following theorem given by the author (but not the other proofs; I am not trying to duplicate a question, but I am trying to understand assertion made by author.)
Theorem 3.8 The greatest common divisor of the integers $a$ and $b$, not both $0$, is the least positive integer that is an integral linear combination of $a$ and $b$.
Method of Proof: Step (1) Let $d$ be the least positive integer that is a linear combination of $a$ and $b$.
Step $2$: We will show that $d|a$ and $d|b$. [I am skipping proof, since my main question is later.]
Step $3$: Let $c$ be a common divisor of $a$ and $b$. We will show that $c|d$. [ Again, I am skipping proof of this.]
After this, author asserts the following:
From Theorem 3.8, we immediately see that the greatest common divisor of $a$ and $b$ can be written as a linear combination of these integers.
Question: I don't understand how he made this assertion from above proof?
More specifically, I thought about proof as follows: $$ \Big{(}d=\mbox{least positive integer, which is integral combination of $a$ and $b$}\Big{)} \,\, \Longrightarrow \,\, \Big{(} d=gcd(a,b)\Big{)} $$ and Steps 1 to 3 give proof of this.
BUT how the author asserts bout the converse of this again from steps 1 to 3, I am not getting.
From Theorem 3.8, in a similar way to how you wrote it $$gcd(a,b)= d = \text{least positive integer that is an integral linear combination of a and b}$$ where the assertion says that $$gcd(a,b)=\text{linear combination of a and b}$$
This is clearly true, as the least positive integer linear combination is a linear combination.