Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Answer is 25 i.e. D. Can someone please explain how?

Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors. For more on number properties check: math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION: How many factors does 36^2 have? A. 2 B. 8 C. 24 D. 25 E. 26

\(36^2=(2^2*3^2)^2=2^4*3^4\) and according to above it'll have (4+1)(4+1)=25 different positve factros, including 1 and 36^2 itself.

Answer: D.

5 seconds approach: 36^2 is a perfect square. # of factors of perfect square is always odd (as perfect square has even powers of its primes then when adding 1 to each and multiplying them as in above formula you'll get the multiplication of odd numbers which is odd). Only answer choice D is odd thus it must be correct.

Answer: D.

Tips about the perfect square: 1. The number of distinct factors of a perfect square is ALWAYS ODD. 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. 4. Perfect square always has even powers of its prime factors.

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Hi Bunuel, can you please clarify when we count multiplies of 1? When do we need to use (1+1)(4+1)(4+1) applicable to this question. Thanks
_________________

Don't understand your question. Can you please elaborate?

I mean 36^2=(2^4)*(3^4), the number of factors is (4+1)(4+1)=25 - this is clear. However 36^2 could be (1^1)(2^4)(3^4), which comes to (1+1)(4+1)(4+1)=50. In which cases in GMAT should I count the powers of 1, which doubles the number of factors? Thank you
_________________

Don't understand your question. Can you please elaborate?

I mean 36^2=(2^4)*(3^4), the number of factors is (4+1)(4+1)=25 - this is clear. However 36^2 could be (1^1)(2^4)(3^4), which comes to (1+1)(4+1)(4+1)=50. In which cases in GMAT should I count the powers of 1, which doubles the number of factors? Thank you

Understood now. The answer is never.

To find the number of factors of a positive integers we should make its prime factorization. Now, 1 is not a prime, thus you shouldn't write it when making prime factorization of an integer.

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

Its been long time coming. I have always been passionate about poetry. It’s my way of expressing my feelings and emotions. And i feel a person can convey...

Written by Scottish historian Niall Ferguson , the book is subtitled “A Financial History of the World”. There is also a long documentary of the same name that the...

Post-MBA I became very intrigued by how senior leaders navigated their career progression. It was also at this time that I realized I learned nothing about this during my...