What is the Smallest Positive Integer that has Exactly 18 Positive GMAT Problem Solving

Question: What is the smallest positive integer that has exactly 18 positive factors?

1. 180
2. 216
3. 240
4. 256
5. None of the above

Answer:

Solution with Explanation:
Approach Solution (1):

Let n be the smallest positive integer that has exactly 18 positive factors. Since 18 = 2 * 3 * = (1 + 1) (2 + 1) (2 + 1), we want \(n=a^1*b^2*c^2\) , where a, b, and c are distinct prime numbers.
Since we want n as small as possible, we want a, b, and c to be the three smallest prime numbers, i.e., 2, 3, and 5.
However,, to make n the smallest, we want the largest prime number to have the smallest exponent and the smallest prime number to have the largest exponent.

Therefore, \(n = 5^1*3^2*2^2 = 5*9*4 = 180\)

Correct Option: A

Approach Solution (2):

18 = 9 * 2 = (8 + 1) * (1 + 1)
i.e. number may be \(2^8*3^1=768\)

OR

18 = 6 * 3 = (5 + 1) * (2 + 1)
i.e. number may be \(2^5*3^2= 288\)

OR

18 = 2 * 3 * 3 = (1 + 1) * (2 + 1) * (2 + 1)
i.e. number may be \(2^2*3^2*5=180\)

i.e. smallest such number must be 180

Correct Option: A

“What is the smallest positive integer that has exactly 18 positive factors”- 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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