The first two nonzero elements of the $A$-module $A /\!/ A(n)$ are the generator $e$ (in degree 0) and $Sq^{2^{n+1}} e$ (in degree $2^{n+1}$): all other elements in this degree and lower are a sum of products of lower squares, as a consequence of the Adem relations. When $n=3$ this bottom class is $Sq^{16}$.
However, Adams showed that these higher squaring operations can be decomposed in terms of secondary cohomology operations: there are "operations" $\Phi_{i,j}$ that are only defined on elements where certain Steenrod operations vanish, and relations that say that for $k > 3$ the element $Sq^{2^k}$ is a linear combination of Steenrod operations applied to $\Phi_{i,j}$. The exact relations aren't important here: the important part is that we get some relation $$ Sq^{2^{n+1}}e = \sum Sq^{I_l} (\Phi_{i_l,j_l} e). $$ The elements $\Phi_{i_l,j_l}$ are all in degrees strictly between 0 and $2^{n+1}$ and hence are forced to be zero, so $Sq^{2^{n+1}} e = 0$ for a contradiction.
The machinery behind this is in Adams' paper "On the nonexistence of elements of Hopf invariant one" (Ann. Math, 1960), doi:10.2307/1970147.
