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What is the smallest positive common difference of a 6-term arithmetic progression consisting entirely of (positive) prime numbers?

are divisibility rules applicable here?

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See this Math Overflow question. In your case, divisibility tells us that the difference must be $\geq$ than $$2\cdot 3\cdot 5=30.$$ –  Eric Naslund Mar 22 '13 at 20:25

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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.

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