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text: '[summary: The **least common multiple (LCM)** of two positive [45h natural numbers] a, b is the smallest natural number that both a and b divide, so for instance LCM(12,10) = 60.]\n\nGiven two positive natural numbers $a$ and $b$, their **least common multiple** $\\text{LCM}(a,b)$ is the smallest natural number divided by both $a$ and $b$. As an example take $a=12, b=10$, then the smallest number divided by both of them is $60$.\n\nThere is an equivalent definition of the LCM, which is strange at first glance but turns out to be mathematically much more suited to generalisation: the LCM $l$ of $a$ and $b$ is the natural number such that for every number $c$ divisible by both $a$ and $b$, we have $l$ divides $c$.\nThis describes the LCM as a [3rc poset least upper bound] (namely the [-3rb] $\\mathbb{N}$ under the relation of divisibility).\n\nNote that for $a$, $b$ given, their product $ab$ is a natural number divided by both of them. The least common multiple $\\text{LCM}(a,b)$ divides the product $ab$ and for $\\text{GCD}(a,b)$ the [-5mw] of $a, b$ we have the formula\n$$a\\cdot b = \\text{GCD}(a,b) \\cdot \\text{LCM}(a,b). $$\nThis formula offers a fast way to compute the least common multiple: one can compute $\\text{GCD}(a,b)$ using the [euclidean_algorithm] and then divide the product $ab$ by this number.\n\nIn practice, for small numbers $a,b$ it is often easier to use their factorization into [4mf prime numbers]. In the example above we have $12=2 \\cdot 2 \\cdot 3$ and $10=2 \\cdot 5$, so if we want to build the smallest number $c$ divided by both of them, we can take $60=2 \\cdot 2 \\cdot 3 \\cdot 5$. Indeed, to compute $c$ look at each prime number $p$ dividing one of $a,b$ (in the example $p=2,3,5$). Then writing $c$ as a product we take the factor $p$ the maximal number of times it appears in $a$ and $b$. The factor $p=2$ appears twice in $12$ and once in $10$, so we take it two times. The factor $3$ appears once in $12$ and zero times in $10$, so we only take it once, and so on.',
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