Is it true that GCD$(\alpha,b)=1$?

Question

Let $d=$GCD$(a,b)$, and $\alpha,\beta\in\mathbb Z$ such that $\alpha\cdot a+\beta \cdot b=d$.
Is it true that GCD$(\alpha,b)=1$?

Answer 1

Here is a counterexample: $$2\cdot4-1\cdot6=2.$$

Answer 2

Express $a=d \cdot a'$ and $b= d \cdot b'$, replace in your equation and see what happens! Be lucky