If $k$ be the minimum number of elements that must be added to the relation $R=\{(1,2),(2,3)\}$ on the set $\{1,2,3\}$ so that it is an equivalence relation then the number of positive divisors of $2k+2$ is________

The given set is $A=\left\{1,2,3\right\}$

The given relation is $\left\{\left(1,2\right),\left(2,3\right)\right\}$

To make the above relation, reflexive, add the ordered pairs $\left(1,1\right),\left(2,2\right),\left(3,3\right)$

To make the above relation, symmetric add the ordered pairs $\left(2,1\right),\left(3,2\right)$

Now the given  relation becomes

$R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right),\left(2,3\right),\left(3,2\right)\right\}$

To make the above relation is transitive: 

$\left(1,2\right),\left(2,3\right)\in R$, $\left(1,3\right)$ should belongs to $R$ so add it

Because $R$ is symmetric add $\left(3,1\right)$ also

Therefore, the complete relation $R$ is $\{(1,1),(2,2),(3,3),(1,2),(2,3),(2,1),(3,2),(1,3),(3,1)\}$

Therefore, the number of ordered pairs added to the relation $R$ to make it equivalence relation is $k=7$

The positive divisors of $2k+2=16$ are $1,2,4,8,16$

Therefore, the number of positive divisors is $\boxed{\mathbf{5}}$