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  text: '[summary: The greatest common divisor of two [45h natural numbers] is the largest number which divides both of them.]\n\nThere are two ways to define the **greatest common divisor** (also known as **greatest common factor**, or **highest common factor**), both equivalent.\n\nThe first definition is as the name suggests: the GCD of $a$ and $b$ is the largest number which divides both $a$ and $b$.\n\nThe second definition is the more "mathematical", because it generalises to arbitrary [3gq rings] rather than just [55j ordered rings].\nThe GCD of $a$ and $b$ is the number $c$ such that $c \\mid a$, $c \\mid b$, and whenever $d \\mid a$ and $d \\mid b$, we have $d \\mid c$.\n(That is, it is the maximal element of the [-3rb] that consists of the divisors of $a$ and $b$, ordered by division.)\n\n# Examples\n\n[todo: show the two different definitions in action and how they prepare]\n\n# Equivalence of the definitions\n\n[todo: prove this]\n\n# Relation to prime factorisations\n\n[todo: algorithm given access to prime factorisations; explain why this is unhelpful]\n\n# Calculating the GCD efficiently\n\n[todo: link to Euclidean algorithm]',
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