Divisibility properties of motivic cohomology

We extend results of Colliot-Th'el`ene and Raskind on the ${\mathcal{K}}_2$-cohomology of smooth projective varieties over a separably closed field $k$ to the 'etale motivic cohomology of smooth, not necessarily projective, varieties over $k$. Some consequences are drawn, such as the degeneration of the Bloch-Lichtenbaum spectral sequence for any field containing $k$.