The Largest 3 Digit Number to Have an Odd Number of Factors GMAT Problem Solving

Question: What is the largest 3 digit number to have an odd number of factors?

1. 625
2. 729
3. 841
4. 943
5. 961

“The Largest 3 Digit Number to Have an Odd Number of 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. The 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:

All positive integers have an even number of positive factors except for integers that are squares of integers
Squares of integers (e.g., 1, 4, 9, 16, 25, 36, etc) have an ODD number of positive factors.
So, the question is "What is the largest 3-digit number that is the square of an integer?"We will try to solve the problem statement now.

30² = 900, so 900 will have an odd number of positive factors
31² = 961, so 961 will have an odd number of positive factors
32² = 1024, so 1024 will have an odd number of positive factors.

Of course, 1024 is a four-digit number.
So, 961 must be the greatest 3-digit number with an ODD number of positive factors.

Correct Answer: E

Approach Solution 2:

Square of prime numbers has 3 factors i.e odd no of factors. Since we are asked to find the largest 3 digit number with odd no of factors.
We will check each option one by one.
Let's start with
Option E . 961

961 = 31^2
31 is a prime number.
So 961 has 3 factors (ODD). Since 961 is the largest among the options, we don't need to look for other options.

Correct Answer: E

Approach Solution 3:

We know that a number greater than 1 will have an odd number. So, an odd number of factors can be considered only if it is a perfect square.
Then, the largest 3-digit perfect square number is 31 x 31 = 961 (since 32 x 32 = 1,024).
Correct Answer: E

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