Here is a sketch proof.  

**Step 1:**  For sensible spaces or spectra (connected, finite type) $X$, $H^*(X;\tau) $ will have exponent $p$ for all the coefficient groups $\tau$ you list exactly when the image of the Bockstein $\beta: H^*(X;\mathbb Z/p) \rightarrow H^{*+1}(X;\mathbb Z/p)$ equals the kernel of $\beta$.  This can be proved by fooling around with the various coefficient sequences defining the Bockstein. Bill Browder developed this idea into the `Bockstein spectral sequence', which he made good use of.

**Step 2:** So now we need to look at $H^*(H\pi;\mathbb Z/p)$, viewed graded vector space with the Bockstein action, and we want to show that this module is `$\beta$--acylic'.  The Kunneth theorem applies, and we can reduce to the cases when $\pi = \mathbb Z$ or $\pi=\mathbb Z/p^m$.   

Now we use the calculations of Serre, et. al.  First of all, $H^*(H\mathbb Z/p;\mathbb Z/p) = A$, the mod $p$ Steenrod algebra.  When written with Serre's admissible basis, one can see that $A$ is $\beta$--acyclic.  (In fact, $A$ is forced by general Hopf algebra theory to be free as a module over the exterior algebra on $\beta$.)
Then $H^*(H\mathbb Z;\mathbb Z/p) = A/A\beta$, which one checks is also $\beta$--acyclic ($\beta$ acts on the left).  Finally, if $m>1$, then $H^*(H\mathbb Z/p^m;\mathbb Z/p) = A/A\beta \oplus \Sigma A/A\beta$, and so is also $\beta$--acyclic.

So this is what Rudyak meant.

The canonical mod 2 reference is  Serre, Jean-Pierre Groupes d'homotopie et classes de groupes abéliens. (French) Ann. of Math. (2) 58, (1953). 258–294.  Odd primes would have been later, but not by much.