How many positive integers less than 50 have exactly 4 distinct odd factors but no even factors? A. 6 B. 7 C. 8 D. 9 E. 10
(A) Any positive integer with exactly 4 odd factors must be either:(i) a product of two odd and distinct prime numbers;or (ii) the cube of an odd prime number.
Let’s look at (i): the two distinct prime factors of the number we want must be members of the 5element set {3, 5, 7, 11, 13}. 3 could be paired with any of the 4 other elements. 5 can be paired with only with one element, number 7. Add up the possibilities for 3 and 5 and you get 5 products of distinct odd prime numbers that are less than 50.
Turning to (ii), only the cube of 3 will be less than 50.
In all then, there are 6 positive integers less than 50 that have 4 odd factors and no even ones. The right answer is choice (A).
Another way to tackle this problem is just to find all the numbers (ideally within less than 3 minutes). 1: 15 (1, 3, 5, 15); 2: 21 (1, 3, 7, 21); 3: 27 (1, 3, 9, 27); 4: 33 (1, 3, 11, 33); 5: 35 (1, 5, 7, 35); 6: 39 (1, 3, 13, 39); 
Can 45 be one of these numbers? It can be divided by 1, 45, 5, and 9.
