Linear combinations of the GCD

Question

How does one properly argue whether, if: $$\{s,t \in \mathbb{Z} | s \cdot a + t \cdot b = gcd(a,b)\}$$

Then there exists another pair of numbers s' and t' satisfying $$s' \cdot a + t' \cdot b = gcd(a,b)$$

I guess the question is equivalent to asking how to determine whether there exists more than one integer linear combination that can express the GCD of two numbers a and b

Answer 1

note that $$sa +tb=d \iff (s+kb)a+(t-ka)b=s’a+t’b=d$$