📐 Bézout

For any integers a and b, there exist integers x and y such that ax + by = gcd(a, b).

Proof: The extended Euclidean algorithm computes x and y by back-substitution. Starting from gcd(a,b) = gcd(b, r_1) = \*s = r_n, each step r_(i-1) = q_i r_i + r_(i+1) can be rearranged to express r_(i+1) ...

From: Disquisitiones Arithmeticae
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