Groups and the Group Axioms · Hard problem

For a positive integer $k$, let $M_k$ be the set of $2\times2$ integer matrices with determinant exactly $k$. Since $\det(AB)=\det(A)\det(B)$, a product of two matrices from $M_k$ always has determinant $k^2$. For how many integers $k$ with $1\le k\le10$ does this force $M_k$ to fail closure under matrix multiplication (i.e. $k^2\ne k$)?