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?

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 […]

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 […]

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 […]

Im trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multipe is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple $q$ of $n,m$ is a multiple of $k$, so that there exists a $g$ so that $q=k·g$. Logically […]

Given $m \in Z$, let $mZ$ denote the set of integer multiples of $m$, i.e.: $m\mathbb Z := \{mk\mid k \in \mathbb Z\}$. Now let $a,b \in \mathbb Z$ with $a,b$ not both $0$. Prove that $aZ \cap bZ = \operatorname{lcm}(a,b)Z$. I am trying to write a proof for this, but I am unsure of […]

Given a list $A$ of $n$ positive integer numbers. We’re gonna play this game: $1 -$ Take randomly $2$ numbers of $A$. $2 -$ Delete this $2$ elements of $A$. $3 -$ Insert in $A$ their gcd and lcm. $4 -$ Go to step $1.$ Prove that after some quantity of steps, $A$ doesn’t change […]

Let $a,b\in \mathbb{N}$ and $m=\operatorname{lcm}(a,b)$. Prove that $\gcd(\frac{m}{a},\frac{m}{b})=1$. Any suggestion how to prove I would appreciate. I have try to it directly but I don’t get anything. Probably I’m doing some dumb mistake.

For $a,b \in \mathbb{N}$, how do I prove: $$lcm(a,a+5)=lcm(b,b+5) \implies a=b$$

I think this has been asked before, but I couldn’t find it on math.SE. I googled it too, but I wasn’t lucky enough to find it there either. So, here’s the problem: Demonstrate that for any $a,b,c \in \mathbb{N}$: $$\displaystyle [a,b,c] = \frac{abc}{(ab,bc,ca)}$$ It’s not very hard to verify this equality once we know the […]

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