Bunuel wrote:
What is the smallest positive integer that has exactly 18 positive factors?
A. 180
B. 216
C. 240
D. 256
E. None of the above
Are You Up For the Challenge: 700 Level QuestionsLet’s work the procedure backwards for finding the number of positive factors of a particular integer.
Let n be the smallest positive integer that has exactly 18 positive factors. Since 18 = 2 x 3 x 3 = (1 + 1)(2 + 1)(2 + 1), we want n = a^1 x b^2 x c^2, where a, b, and c are distinct prime numbers. Since we want n as small as possible, we want a, b, and c to be the three smallest prime numbers, i.e., 2, 3, and 5. However, to make n the smallest, we want the largest prime number to have the smallest exponent and the smallest prime number to have the largest exponent. Therefore, n = 5^1 x 3^2 x 2^2 = 5 x 9 x 4 = 180.
Answer: A
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