Let $a, b \in \mathbb{Z}^{*}$, Bézout's identity states that : $$\gcd(a, b) = d \implies \exists (u_0, v_0) \in \mathbb{Z}^{2} \ : \ au_0+bv_0=d$$ And the set of solutions of $au+bv=d$ is : $$S=\left\{\left(u_0 + k \frac{b}{d}, v_0 - k\frac{a}{d}\right) \ \middle| \ k \in \mathbb{Z} \right\}$$