The number of positive integers less than 2009 that have exactly three prime factors, not necessarily distinct, such that each prime factor is greater than 10 is _____________

Correct Answer: 3

Solution

The smallest primes above 10 are 11, 13 and 17. The products of three of them below 2009 are $$11^3 = 1331$$, $$11^2 \times 13 = 1573$$ and $$11 \times 13^2 = 1859$$. Every other product, such as $$11^2 \times 17 = 2057$$ or $$11 \times 13 \times 17 = 2431$$, exceeds 2009, so there are 3 such numbers.

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