Pumping lemma: the set of strings of 0s and 1s such that when interpreted as an integer, that integer is prime
I believe what you are missing is Fermat's Little Theorem: If $p$ is a prime and $p$ does not divide $a$, then $a^{p-1}\equiv 1 \mod p$. 

I'll give an example application in base 10. Let's assume that 2039 is the prime to which the pumping lemma is applied and the decomposition is 2 03 9. That is the pumping lemma would imply that all strings of the form $2(03)^n9$ denote prime numbers. At least it is true for $n\in\{0,1,2\}$. But without further calculation I know that it fails for $n=28$: The number $N=2030303030303030303030303030303030303030303030303030303039$ is divisible by $29$, which is the number at $n=0$. To see this, observe that 
$N = 20\cdot 100^{28} + \sum\limits_{0\leq i&lt; 28}30\cdot 100^i + 9
= 20\cdot 100^{28} + 30\cdot\frac{100^{28}-1}{100-1} + 9$ and apply Fermat's theorem. 
Here, we also use that $29$ does not divide $100-1$. 

I hope from this you are able to solve the exercise.