Question

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]

Hint:

To determine the number of factors of a number, consider its prime factorization. If the number is the square of a prime, it will have exactly 3 factors.

Answer 1

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.