Let $k$ be an algebraically closed field, let $I, J \subseteq k[x_1,\dots, x_n, y_1,\dots, y_m]$ be ideals, and let $i,j$ be the minimum number of generators of $I$ and $J$, respectively. I have two questions:

Is the minimum number of generators of the product ideal $IJ$ at least $\max\{i,j\}$?

If $I$ is generated by polynomials that depend only on $x_1,\dots, x_n$ and $J$ is generated by polynomials that depend only on $y_1,\dots, y_m$, then is the minimum number of generators of the product ideal $IJ \subseteq k[x_1,\dots, x_n, y_1,\dots,y_m]$ equal to $i j$?

Question two is a repost from MSE.