Articles of least common multiple

Existence of the least common multiple in a Unique Factorization Domain

Let $x,y \in R$, where $R$ is a unique factorization domain. Let $P$ be the set of all representatives in each class of associate irreducible elements of $R$. Then, suppose $x,y\in R$ are nonzero, nonunit elements (First Question: How would this proof change if either $x$ or $y$ is zero or is a unit of […]

Ternary generalizations of $\,\gcd(n,m)\,{\rm lcm}(n,m) = nm$

$\gcd(n,m)\,{\rm lcm}(n,m) = nm.\,$ Can this theorem work with 3 integers? And how to prove it? I tried doing this with 2 integers n,m , but I can’t figure out how to do it with 3.

Least common multiple of sequence of integers

Let $a_n$ be a strictly increasing sequence of natural numbers such that $$\lim_{n\to\infty}\frac{a_n}{2^n}=0.$$ Now, define $$ l_n=\frac{lcm(a_1,a_2,…,a_{n-1})}{a_n}.$$ My question is: From the definition of $a_n$, can we say that $l_n\to\infty$? While I haven’t been able to find any previous work answering this question, a similar question proves that the least common multiple of a random […]

Distribution and expected value of a random infinite series $\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$

Can we find the distribution and/or expected value of $$S=\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$$ where $r_n$ is a uniformly distributed random integer, $r_n \in [1,n]$ and $\text{lcm}$ is the least common multiple? Or maybe an estimate is possible? Some boundaries are easy to see. For example: $$1<S \leq \sum_{n \geq 1} \frac{1}{n^2}= \frac{\pi^2}{6} \approx 1.645$$ […]

LCM of binomial coefficients and related functions

I know about the following identity: $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, … {n \choose n} \right) = \frac{\text{lcm}(1, 2, … n+1)}{n+1}$$ 1) Is there any method to find $$\displaystyle \text{lcm} \left( {n \choose 0}, {n \choose 1}, … {n \choose k} \right)$$ for any $k \le n$ without evaluating all the […]

Find all pairs of positive integers that add up to $667$ and their $\frac{\text{lcm}}{\text{gcd}} =120$

Let $x,y$ be positive integers, $x<y$, and $x+y=667$. Given that $\dfrac{\text{lcm}(x,y)}{\text{gcd}(x,y)}=120,$ find all such pairs $(x,y)$. The only way I can think of solving this is trying all possibilities where one number is odd and the other even, and testing them all. Using this, I found one solution, $(115,552)$, but I’m wondering if there is […]

Cancellative Abelian Monoids

Is there an example of cancellative Abelian monoid $M$ in which we may find two elements $x$ and $y$ such that they have a least common multiple but not a greatest common divisor?

Positive Integers Equation

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with the least common multiple, but other than that, the textbook gave no hints really and I’m really not sure about how to approach it. So anyway, here the problem […]

Can the identity $ab=\gcd(a,b)\text{lcm}(a,b)$ be recovered from this category?

Define the category $\mathcal{C}$ as follows. The objects are defined as $\text{Obj}(\mathcal{C})=\mathbb{Z}^+$, and a lone morphism $a\to b$ exists if and only if $a\mid b$. Otherwise $\text{Hom}(a,b)=\varnothing$. It can be shown that $\gcd(a,b)$ is a product in $\mathcal{C}$ and $\text{lcm}(a,b)$ is a coproduct in $\mathcal{C}$. My question is: Can we recover the identity $$ab=\gcd(a,b)\text{lcm}(a,b)$$ from […]

Solve for $A,B$: $\mathrm{LCM}(A,B)=168$, $\mathrm{HCF(A,B)}=12$

The highest common factor and the lowest common multiple of two numbers $A$ and $B$ are $12$ and $168$ respectively. Find the possible values of $A$ and $B$ with the exception of $12$ and $168$. Help me with this equation please. I kind of get it but I don’t understand the concept behind it… I […]