smc

Since the sum of the digits of $N$ is $4$ and none of the digits are $0$, $N$'s digits must the elements of one of the sets $\{1,1,1,1\},\{1,1,2\}$, $\{2,2\}$, $\{1,3\}$, or $\{4\}$. In the first case, $N=1111=101\cdot 11$ so this is a counter example. In the second case, $N=112$ is excluded for being even. With $N=121=11^2$ we have a counterexample. We can check $N=211$ by trial division, and verify it is indeed prime. In the third case, $N=22$ is excluded for being even. In the fourth case, both $N=13$ and $N=31$ are prime. In the last case $N=4$ is excluded for being even. This gives two counterexamples and the answer is $\boxed{2}$