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Bezout’s Identity.md

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+#Math #NT
+
+# Statement
+
+Let $x \in \mathbb{Z}$, $y \in \mathbb{Z}$, $x \neq 0$, $y \neq 0$, and $g = \gcd(x, y)$. Bezout's Identity states that $\alpha \in \mathbb{Z}$ and $\beta \in \mathbb{Z}$ exists when:
+
+$$
+\alpha x + \beta y = g
+$$
+
+Furthermore, $g$ is the least positive integer able to be expressed in this form.
+
+# Proof
+
+## First Statement
+
+Let $x = gx_1$ and $y = gy_1$, and notice $\gcd(x_1, y_1) = 1$ and $\operatorname{lcm} (x_1, y_1) = x_1 y_1$.
+
+Since this is true, the smallest integer $\alpha$ for $\alpha x_1 \equiv 0 \mod y$ is $a = y_1$.
+
+For all integers $0 \leq a, b < y_1$, $ax_1 \not\equiv bx_1 \mod y$. (If not, we get $|b - a| > y_1$, which is contradictory). Thus, by pigeonhole principle, there exists $\alpha$ such that $\alpha x_1 \equiv 1 \mod y_1$.
+
+Therefore, there is an $\alpha$ such that $ax_1 - 1 \equiv 0 \mod y_1$, and by extension, there exists an integer $\beta$ such that:
+
+$$
+\alpha x_1 - 1 = -\beta y_1 \\
+\alpha x_1 + \beta y_1 = 1
+$$
+
+By multiplying by $g$:
+
+$$
+\alpha x + \beta y = g
+$$
+
+## Second Statement
+
+To prove $g$ is minimum, let’s consider another positive integer $g\prime$:
+
+$$
+\alpha\prime x + \beta\prime y = g\prime
+$$
+
+Since all values are a multiple of $g$:
+
+$$
+0 \equiv \alpha \prime x + \beta \prime x \mod g \\
+0 \equiv g\prime \mod g
+$$
+
+Since $g$ and $g\prime$ are positive integers, $g\prime \geq g$.