How Many Positive Factors do 180 and 96 have in Common? GMAT Problem Solving

Question: How many positive factors do 180 and 96 have in common?

1. 6
2. 12
3. 16
4. 18
5. 24

Solution and Explanation:

Approach Solution (1):

The question talks about finding the common positive factors of two numbers. The common factors should have its prime numbers in the power which is present in both the original numbers. Hence our first step would be to prime factorize the original numbers
Once we prime factorize the number we can look at the common prime factors of the number along with their powers to find out the common factors
Prime factorizing 180 and 96 would give us 180 = \(2^2*3^2*5\) and \(96=2^5*3\)

From the above prime factorization of 180 and 96, we can analyze that the common factors of 180 and 96 would come from a combination of \(2^1 and 3^1\).
Thus there are 3 * 2 = 6 possible combinations i.e. 6 common factors of 180 and 96
The common factors would be 1, 2, 3, 4, 6 and 12 which are the factors of HCF (180, 96) = 12

Correct Option: A

Approach Solution (2):

The number of common factors will be same as the number of factors of the Highest Common Factor (HCF)
HCF of 180 and 96 = 12
Number of factors of 12 = 6

Correct Option: A

Approach Solution (3):

To find the number of common factors of two integers:

(1) Find the greatest common divisor (GCD) of the two integers;
(2) Find the number of factors of that GCD.

GCD of 180 and 96 is 12 = \(2^2*3\)

The number of factors of 12 is (2 + 1) (1 + 1) = 6

Correct Option: A

“How many positive factors do 180 and 96 have in common?”- 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 amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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