How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.) [Official GMAT-2018]

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Step 1: Factorization of integers between 1 and 16. The number of factors of a number is determined by its prime factorization. A number will have exactly 3 factors if it is a square of a prime number. Step 2: Check the numbers. - 1 has 1 factor. - 2, 3, 5, 7, 11, and 13 are prime numbers, so they have 2 factors each. - 4 has 1, 2, 4 factors. - 6 has 1, 2, 3, 6 factors. - 8 has 1, 2, 4, 8 factors. - 9 has 1, 3, 9 factors. - 10 has 1, 2, 5, 10 factors. - 12 has 1, 2, 3, 4, 6, 12 factors. - 14 has 1, 2, 7, 14 factors. - 15 has 1, 3, 5, 15 factors. - 16 has 1, 2, 4, 8, 16 factors. Step 3: Conclusion. The numbers 4 and 9 have exactly 3 factors, so the answer is 2.

LIST-I (Infinite Series)	LIST-II (Nature of Series)
(A) \( 12 - 7 - 3 - 2 + 12 - 7 - 3 - 2 + \dots \)	(II) oscillatory
(B) \( 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots \)	(IV) conditionally convergent
(C) \( \sum_{n=0}^{\infty} \left( (n^3+1)^{1/3} - n \right) \)	(I) convergent
(D) \( \sum_{n=1}^{\infty} \frac{1}{n \left( 1 + \frac{1}{n} \right)} \)	(III) divergent

LIST-I (Set)	LIST-II (Supremum/Infimum)
(A) \( S = \{2, 3, 5, 10\} \)	(III) Sup S = 10, Inf S = 2
(B) \( S = (1, 2] \cup [3, 8) \)	(IV) Sup S = 8, Inf S = 1
(C) \( S = \{2, 2^2, 2^3, \dots, 2^n, \dots\} \)	(II) Sup S = 5, Inf S = -5
(D) \( S = \{x \in \mathbb{Z} : x^2 \le 25\} \)	(I) Inf S = 2

How many integers between 1 and 16, inclusive, have exactly 3 different positive integer factors? (Note: 6 is NOT such an integer because 6 has 4 different positive integer factors: 1, 2, 3, and 6.) [Official GMAT-2018]

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