How Many Integers Between 1 and 16 Inclusive Have Exactly 3 Different Positive Integer Factors GMAT Problem Solving

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. 1
2. 2
3. 3
4. 4
5. 5

“How Many Integers Between 1 and 16 Inclusive Have Exactly 3 Different Positive Integer Factors GMAT Problem Solving”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

This method can be used for quick answer-

The prime numbers can be immediately ruled out because they have two factors.Now, what we have are the following numbers: 1, 4, 6, 8, 9, 10, 12, 14, 15, 16
Furthermore remove numbers 1 and all the numbers that are above 10
Now what we are left with are numbers: 4, 6, 8, 9
Furthermore remove numbers 6 and 8 since they have 4 factors.
Now what we are left with are numbers: 4 and 9 and they have exactly 3 positive factors.
There are 2 integers between 1 and 16
The Answer is B which is 2.

Approach Solution 2:

Lets try a different arithmetic approach to find integers that are between 1 and 16,and are inclusive, and have exactly 3 different positive integer factors

The prime numbers (1, 2, 3, 5, 7, 11, 13) have only two positive integer factors, themselves and 1, so now let us just analyze non-prime numbers
The positive integer factors of:
4 are 1, 2, 4 = 3
6 are 1, 2, 3, 6 = 4
8 are 1, 2, 4, 8 = 4
9 are 1, 3, 9 = 3
10 are 1, 2, 5, 10 = 4
12 are 1, 2, 3, 4, 6, 12 = 6
14 are 1, 2, 7, 14 = 4
15 are 1, 3, 5, 15 = 4
16 are 1, 2, 4, 8, 16 = 5
Now as it can be seen above, 4 and 9 are two numbers with positive integers.
Therefore,
The Answer is B which is 2.

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