What is the smallest positive common difference of a 6-term arithmetic progression consisting entirely of (positive) prime numbers?
are divisibility rules applicable here?
Yes, divisibility rules are important here. Clearly the difference must be even as all primes (except $2$) are odd. The difference must be a multiple of $3$ because otherwise two of the numbers in the progression will be multiples of $3$. Carry on. Since primes get less common as the numbers get larger, you should try starting small.