How Many Numbers Between 1 and 1000, Inclusive Have an Odd Number of Factors? GMAT Problem Solving

Question: How many numbers between 1 and 1000, inclusive have an odd number of factors?

1. 10
2. 25
3. 31
4. 64
5. 128

‘How many numbers between 1 and 1000, inclusive have an odd number of factors? ' - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “The Official Guide for GMAT Reviews”. 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.

Solution and Explanation:

Approach Solution 1:

It is asked in the question that how many numbers between 1 and 1000, inclusive have an odd number of factors. This is a question from the number theory.
It should be noted that only square numbers are found to have the odd number of factors.
The question is ultimately asking to find out the number of squares which come in the range of 1-1000.
To find the number of squares which lie in the range 1-1000
Take square root of 1000
\(\sqrt{1000}\)= 31.6
We get 31.6 Therefore 31^2 will always be less than 1000.
Therefore there are 31 squares which lie in the range 1-1000
The correct answer will be option C.

Correct Answer: C

Approach Solution 2:
It is asked in the question that how many numbers between 1 and 1000, inclusive have an odd number of factors. This is a question from the number theory.
There is a property that an integer with odd number of factors is a perfect square of an integer and only has even exponents in its prime factorization.
Let n = \(p^aq^br^c\), then n has (a+1)(b+1)(c+1) factors.
Where p,q,r are prime numbers.
A,b,c are positive integers. So (a+1),(b+1),(c+1) must be odd numbers in order for n to have odd number of factors. This implies that a,b,c are even numbers and n is a perfect square. To find the number of
squares which lie in the range 1-1000
Take square root of 1000
\(\sqrt{1000}\)= 31.6
There are total of 31 perfect squares in the range 1-1000.
Therefore the correct answer will be option C.

Correct Answer: C

Suggested GMAT Problem Solving Questions

• If x and y are positive integers and(17)x∗(18)12=(18)18y(17)x∗(18)12=(18)18y, then what is the value of y-x? GMAT Problem Solving
• At Springfield High, Three-Fourths of the Male Students and Half of the Female Students Speak a Foreign Language GMAT Problem Solving
• What is the Number of Integers from 1 to 1000, Inclusive that are not Divisible by 11 or by 35? GMAT Problem Solving
• If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m? GMAT Problem Solving 
• If Ben Were to Lose the Championship, Mike would be the Winner GMAT Problem Solving
• A Train Travelling at a Certain Constant Speed takes 30 seconds GMAT Problem Solving
• A Conference Room is Equipped with a Total of 45 Metal or Wooden Chairs. GMAT Problem Solving 
• A Welder Received an Order to Make a 1 Million Litre Cube-Shaped Tank. GMAT Problem Solving 
• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving 
• If 12 Ounces of a Strong Vinegar Solution are Diluted with 50 Ounces of Water to Form a Three-Percent Vinegar Solution GMAT Problem Solving 
• A Contractor Estimated that his 10-Man Crew Could Complete the Construction in 110 Days if There Was no Rain. GMAT Problem Solving 
• A Circle is Inscribed in a Square with the Diagonal of 4 Centimeters. GMAT Problem Solving 
• At a Dog Competition, a Dog is Awarded 10 Points if it Runs Through 4 Pipes, Makes 10 Jumps, and Walks on 2 Beams. GMAT Problem Solving 
• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours. GMAT Problem Solving 
• What is the Value of k if the Sum of Consecutive Odd Integers From 1 to k Equals 441? GMAT Problem Solving 
• A Circle with a Radius R is Inscribed into a Square with a Side K. GMAT Problem Solving 
• If 53 Students are Enrolled in Both the CS103 (Algorithms and Data Structures) And M101 (Mathematics for Computer Science) Classes GMAT Problem Solving 
• The square of 5√252 =? GMAT Problem Solving 
• The Population of the Bacteria Colony Doubles Every Day. If It was Started 9 Days Ago with 2 Bacteria and each Bacteria Lives for 12 Days GMAT Problem Solving 
• If tu=xytu=xyand ty=uxty=ux Where t, u, x, and y are Non-Zero Integers, Which of the Following is True? GMAT Problem Solving