How many positive integers less than 30 are co-prime to 30?

30 = $2\times 3\times 5$, so a number is co-prime to it only if it avoids the factors 2, 3 and 5. Of the numbers 1 to 29 that leaves 1, 7, 11, 13, 17, 19, 23 and 29 — eight in all. (Counting them as $30\times 1/2\times 2/3\times 4/5=8$ gives the same result.)