TITLE: In the ring of even integers , do 4 and 6 have a lcm and gcd? If they have , what are they?
QUESTION [5 upvotes]: I have a solution, but I have a doubt in it.
regarding gcd :
2 is not gcd for 4 and 6 because 2 does not divides 6 , as it gives 3 which does not belongs to set of even integers ring.
regarding lcm : (i have doubt)
the solutions shows 12 is not lcm because 4|12 (= 3) which does not belongs to set of even integers ring but then why not 24 is a solution to lcm of 4 and 6 in a set of even integers??

REPLY [6 votes]: Let $E$ be the (nonunital) ring of even integers. A common divisor of $4$ and $6$ in $E$ is also a common divisor in the integers, so the only candidate is $2$; however, as you observe, this is not a divisor of $6$ in $E$. Therefore the two elements have no common divisor.
Similarly, a common multiple in $E$ is also a common multiple in the integers, so we have to look to numbers of the form $12x$; since $12x/4=3x$, we need $x$ to be even. The candidate for the lcm is thus $24$. Is it?
Note that “least common multiple” means “a common multiple that divides every other common multiple” (the standard order relation in the integers is not considered).
Now try $72$.
Can any other common multiple be the lowest common multiple?