How many divisors of 900 are smaller than 30 yet do not divide 30?

Because $900=2^2\times 3^2\times 5^2$ it has $3\times 3\times 3=27$ divisors, and since $900=30^2$ they pair off around 30: exactly 13 lie below, one is 30 itself and 13 lie above. The 13 below are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20 and 25. Of these, the ones that DO divide 30 are 1, 2, 3, 5, 6, 10 and 15 — seven of them, being all the divisors of 30 apart from 30. Removing them leaves 4, 9, 12, 18, 20 and 25, so the answer is 6. The trap 13 forgets to remove anything; 7 and 8 count the divisors of 30 rather than the survivors.