The purpose of this paper is to study the p-part of motivic cohomology and algebraic K-theory in characteristic p (we use higher Chow groups as our definition of motivic cohomology). The main theorem states that for a field k of characteristic p, H^i(k,Z(n)) is uniquely p-divisible for i\not=n. This implies that the natural map K_n^M(k) --> K_n(k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and K_n(k) is p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K_n(X,Z/p^r)=0 for n>dim X. Another consequence is Gersten's conjecture with mod p coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch's cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except the vanishing conjecture.