GMAT Practice Question: How many integers between 1 and 16, inclusive, have exactly 3 different positive integer...
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.)
- 1
- 2
- 3
- 4
- 6
Topics: number theory, factoring, prime factorization, prime numbers
Solution
Step 1
We observe that an integer has exactly three positive factors only when it is the square of a prime.
n = p^2 \text{where } p \text{ is prime}
Step 2
We determine all prime values of p for which p² is at most 16, by square rooting the inequality
p^2 \le 16 p \le 4 \text{primes } p = 2,3
Step 3
We list the squares of these primes and count how many lie between 1 and 16.
2^2 = 4 3^2 = 9 \text{number of such integers} = 2
Answer
2
