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text: '[summary: Bézout's theorem states that if $a$ and $b$ are integers, and $c$ is an integer, then the equation $ax+by = c$ has integer solutions in $x$ and $y$ if and only if the [-5mw] of $a$ and $b$ divides $c$.]\n\nBézout's theorem is an important basic theorem of number theory.\nIt states that if $a$ and $b$ are integers, and $c$ is an integer, then the equation $ax+by = c$ has integer solutions in $x$ and $y$ if and only if the [-5mw] of $a$ and $b$ divides $c$.\n\n# Proof\n\nWe have two directions of the equivalence to prove.\n\n## If $ax+by=c$ has solutions\n\nSuppose $ax+by=c$ has solutions in $x$ and $y$.\nThen the highest common factor of $a$ and $b$ divides $a$ and $b$, so it divides $ax$ and $by$; hence it divides their sum, and hence $c$.\n\n## If the highest common factor divides $c$\n\nSuppose $\\mathrm{hcf}(a,b) \\mid c$; equivalently, there is some $d$ such that $d \\times \\mathrm{hcf}(a,b) = c$.\n\nWe have the following fact: that the highest common factor is a linear combination of $a, b$. ([hcf_is_linear_combination Proof]; this [extended_euclidean_algorithm can also be seen] by working through [euclidean_algorithm Euclid's algorithm].)\n\nTherefore there are $x$ and $y$ such that $ax + by = \\mathrm{hcf}(a,b)$.\n\nFinally, $a (xd) + b (yd) = d \\mathrm{hcf}(a, b) = c$, as required.\n\n# Actually finding the solutions\n\nSuppose $d \\times \\mathrm{hcf}(a,b) = c$, as above.\n\nThe [-extended_euclidean_algorithm] can be used (efficiently!) to obtain a linear combination $ax+by$ of $a$ and $b$ which equals $\\mathrm{hcf}(a,b)$.\nOnce we have found such a linear combination, the solutions to the integer equation $ax+by=c$ follow quickly by just multiplying through by $d$.\n\n# Importance\n\nBézout's theorem is important as a step towards the proof of [5mh Euclid's lemma], which itself is the key behind the [5rh].\nIt also holds in general [5r5 principal ideal domains].',
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