TITLE: Can a nontrivial spectrum smash to zero with $K$-theory? 
QUESTION [6 upvotes]: Let $E $ be a (possibly nonconnective) spectrum. Suppose $E \wedge K = 0$ (where $K$ is complex $K$-theory). Does it follow that $E = 0$?

REPLY [10 votes]: Another type of example: Let $E$ be the mod $p$ homology spectrum. The integral homology groups of the periodic $K$-theory spectrum are rational vector spaces, i.e. the integral homology groups of the mod $p$ spectrum are zero.

REPLY [8 votes]: Sure.  Smashing a based space with a spectrum is equivalent to smashing
its suspension spectrum with that spectrum.  So it suffices to give a 
nontrivial space whose reduced $K$-homology is trivial.  A classical 
example due to Luke Hodgkin is $Coker J$.  See Hodgkin, Luke; Snaith, Victor.
The K-theory of some more well-known spaces. 
Illinois J. Math. 22 (1978), no. 2, 270–278.