2007/08 AHSMC Part 1 Problem 5

The integers from $11$ to $30$ include six primes, namely $11$, $13$, $17$, $19$, $23$ and $29$. In the ten odd numbers among any twenty consecutive integers each at least $7$, at least three are multiples of $3$ and exactly two are multiples of $5$, but at most one can be a multiple of $15$. Hence the maximum is indeed six. The answer is (C).