Vector fields on the $5$-sphere What is the maximum number of smooth, linearly-independent vector fields that can exist on the $5$-sphere?

1) There certainly exists at least $1$ smooth linearly independent vector field on $S^5$ (i.e. nonzero at each $x\in S^5$), namely $x=(x_1,x_2,x_3,x_4,x_5,x_6)\mapsto (x_2,-x_1,x_4,-x_3,x_6,-x_5)$ . 
2) There do not even exist $2$ linearly independent continuous vector fields on $S^5$.
This is proved  in Steenrod's The Topology of Fibre Bundles, published in 1951 (before $K$-theory was invented): Theorem 27.11, page 142.
No Chern classes are  used either.
The only tool used is (rather advanced) homotopy theory, which is developed in Part II of the book.   
Another reference
As mentioned by @user10354138, Adams has reproved  the result above, and much more !, using K-theory (a freshly invented technique in 1961) in his article Vector Fields on Spheres
