Let $E $ be a (possibly nonconnective) spectrum. Suppose $E \wedge K = 0$ (where $K$ is complex $K$-theory). Does it follow that $E = 0$? 



Can a nontrivial spectrum smash to zero with $K$-theory? 
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. 


