$X$ is the number of different positive integers that divide the chosen number without remainder. For example, if $6$ is chosen , then the value of $X$ is $4$ since $1,2,3,6$ each divide $6$ without remainder.
First construct a table with two rows listing integers between 1 and 10 and the divisors of each integer (which I have already done).
Second use this table to construct another take with two rows listing the unique values $X$ assumes (i.e listing the range of $X$) and the probability $p(x) = P(X=x)$ for each value.
Next, use your answer from the first to parts of find $E(X)$ and $Var(X)$.