$A = \{n | 2 \le n \le 12\}$
$D$ is relation on $A$ defined: $(a,b) \in D$ iff $b \pmod{a} \equiv 0$.
I need to find the elements which are minimal regarding to D.
What I think to do is to find all Equivalence Class and then take the minimal in each Equivalence Class. But here I think I miss something.
2 and 12 need to be in the same Equivalence Class, since 12 mod 2 === 0 2 and 10 need to be in the same Equivalence Class, since 10 mod 2 === 0
but each element can appear only in one Equivalence Class and so 2 => it mean that 2,10,12 need to be in the same class but 12 mod 10 !== 0
What I miss here? thanks
Hint:
To say that $b\equiv0\pmod a$ is equivalent to saying that $a\mid b$. So, your relation is that $(a,b)\in D$ if and only if $a\mid b$.
So, what sort of element is minimal in this setup? Being minimal means that a number isn't divisible by any number in $\{2,3,\ldots,12\}$ other than itself.