Let $R_1, R_2$ and $R_3$ be a relations in $\mathbb{N}$, where $\mathbb{N} = \{0, 1, 2, \ldots\}$. $m|n$ means $m$ divides $n$. For which positive integers $k$, these relations are equivalence relations? Please comment, if my answers are correct. Thanks in advance.
a) $aR_1b$ if and only if $k | (a+b)$.
In this case, I think $k$ can only be $1$ or $2$. Relation is even not reflexive for any other case.
b) $aR_2b$ if and only if $k | (a-b)$.
Now relation is reflexive, symmetric and transitive for any $k > 0$. First two are obvious, it's transitive because for $k' > 0$ we have $a-b=k' \cdot m$ and $b-c = k' \cdot m'$ implies $a-c = k'(m+m')$.
c) $aR_3b$ if and only if $x-y=k$.
It can be reflexive only for $k=0$, but $k$ must be positive integer, so there is no such $k$ that $R_3$ is equivalence relation.