GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Aug 2019, 22:34 ### GMAT Club Daily Prep

#### 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. ### Request Expert Reply # How many factors does 36^2 have?

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 462
Location: United Kingdom
Concentration: International Business, Strategy
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
How many factors does 36^2 have?  [#permalink]

### Show Tags

2
23 00:00

Difficulty:   15% (low)

Question Stats: 68% (00:46) correct 32% (01:05) wrong based on 339 sessions

### HideShow timer Statistics

How many factors does 36^2 have?

A. 2
B. 8
C. 24
D. 25
E. 26

_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730

Originally posted by enigma123 on 22 Jan 2012, 16:48.
Last edited by Bunuel on 31 Jul 2012, 02:53, edited 1 time in total.
Edited the question and added the OA.
##### Most Helpful Expert Reply
Math Expert V
Joined: 02 Sep 2009
Posts: 57083
Re: Factors of 36^2  [#permalink]

### Show Tags

25
17
enigma123 wrote:
How many factors does 36^2 have?

2 8 24 25 26

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.

Hope it helps.
_________________
##### General Discussion
Director  Status: Done with formalities.. and back..
Joined: 15 Sep 2012
Posts: 581
Location: India
Concentration: Strategy, General Management
Schools: Olin - Wash U - Class of 2015
WE: Information Technology (Computer Software)
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

valerjo79 wrote:
How many factors does 36^2 have?

Does anyone see a fast solution?

Thanks

2
8
24
25
26

$$36^2 = (2^2 * 3^2)^2 = 2^4*3^4$$

Hence number of factors = ($$4+1) * (4+1) = 25$$.

Ans D it is!
_________________
Lets Kudos!!! Black Friday Debrief
Intern  Joined: 21 Oct 2012
Posts: 23
GMAT Date: 01-19-2013
Re: Factors of 36^2  [#permalink]

### Show Tags

Bunuel wrote:
enigma123 wrote:
How many factors does 36^2 have?

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
_________________
MGMAT1 - 610
MGMAT2 - 670
MGMAT3 - 640

OMG
Math Expert V
Joined: 02 Sep 2009
Posts: 57083
Re: Factors of 36^2  [#permalink]

### Show Tags

fukirua wrote:
Bunuel wrote:
enigma123 wrote:
How many factors does 36^2 have?

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?
_________________
Intern  Joined: 21 Oct 2012
Posts: 23
GMAT Date: 01-19-2013
Re: Factors of 36^2  [#permalink]

### Show Tags

Bunuel wrote:

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
_________________
MGMAT1 - 610
MGMAT2 - 670
MGMAT3 - 640

OMG
Math Expert V
Joined: 02 Sep 2009
Posts: 57083
Re: Factors of 36^2  [#permalink]

### Show Tags

fukirua wrote:
Bunuel wrote:

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.

Hope it's clear.
_________________
Manager  Joined: 15 Jan 2014
Posts: 88
Concentration: Healthcare, Strategy
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

I remember reading a rule in MGMAT that any perfect square will have an odd number of factors. 25 is the only odd answer choice. Is that right?
Math Expert V
Joined: 02 Sep 2009
Posts: 57083
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

1
4
lmariesf wrote:
I remember reading a rule in MGMAT that any perfect square will have an odd number of factors. 25 is the only odd answer choice. Is that right?

Yes, it is. Check my solution here: how-many-factors-does-36-2-have-126422.html#p1032696

Tips about positive perfect squares:

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.

Hope this helps.
_________________
Target Test Prep Representative D
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 7393
Location: United States (CA)
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

enigma123 wrote:
How many factors does 36^2 have?

A. 2
B. 8
C. 24
D. 25
E. 26

We can use the rule of adding 1 to each exponent of each unique prime and then multiplying our values together:

36^2 = (9 x 4)^2 = (3^2 x 2^2)^2 = 3^4 x 2^4

(4 + 1)(4 + 1) = 5 x 5 = 25

Alternate solution:

An interesting fact about perfect squares greater than 1 is that they always have an odd number of factors. For example, the factors of 4 are 1, 2, and 4 (a total of 3 factors) and the factors of 100 are 1, 2, 5, 10, 20, 50, and 100 (a total of 7 factors). Since 36^2 is a perfect square, it should have an odd number of factors. The only odd number in the answer choices is 25.

Answer: D
_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Intern  B
Joined: 19 Jun 2017
Posts: 4
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

Isn't it supposed to be how many positive factors does 36^2 have?
Otherwise, the answer should be 50.
(-36)*(-36)= 36^2 as well. ect
Math Expert V
Joined: 02 Sep 2009
Posts: 57083
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

mahagmat wrote:
Isn't it supposed to be how many positive factors does 36^2 have?
Otherwise, the answer should be 50.
(-36)*(-36)= 36^2 as well. ect

A factor is a POSITIVE divisor.
_________________
Intern  B
Joined: 19 Jun 2017
Posts: 4
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

Why is that ?
For example, 36= 36 x 1 ( 36 and 1 are factors of 36)
But, 36= -36 x -1 ( -36 and -1 are also factors of 36)
I came across a question once that said : What is the sum of all factors of X, the answer was 0, which for me was corehent.

On magoosh, the mention both negative and positive factors :
"For the GMAT: The same rules in the GRE apply to the GMAT in this case. Once again, the factors and multiples of an integer include both the positive and negative integers/multiples.
Once again, we haven't seen an official question that involved negative factors or multiples, but it doesn't mean that they don't exist. However, like the GRE, the questions will also clearly specify "positive factors" or "positive integers". :D"

Can you please explain how do you only consider positive factors in this case?
Thank you.
VP  D
Joined: 09 Mar 2016
Posts: 1258
How many factors does 36^2 have?  [#permalink]

### Show Tags

Quote:

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.

Hope it helps.

Hello niks18, could you please advice regarding Bunuels tips about the perfect square. please see below and advice what i missed or did wrong, or counted incorrectly many thanks!

lets take 36. here is prime factorization of 36 (2*18) ---> (18) ---> (2*9) ---> (9) --->(3*3)

$$Tips about the perfect square$$

1. The number of distinct factors of a perfect square is ALWAYS ODD. lets take 36 in all cases to test your tips:) - it has 1, 2, 3, 9 = 4 as distinct factors, no but it is EVEN 2. The sum of distinct factors of a perfect square is ALWAYS ODD. 36 - so it has 1, 2, 3, 9 as distinct factors hence 1+ 2+3+9 = 15 is it correct ?

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. let me break down this sentence into two clauses:)

A perfect square ALWAYS has an ODD number of Odd-factors ---> odd numbers are 1, 3, 9 so total 3 odd numbers

A perfect square ALWAYS has an EVEN number of Even-factors ---> 2, 8 so in total 2 numbers is corrrect but how about 6 ! ?? or is my prime factorization incorrect ??

4. Perfect square always has [b]even powers of its prime factors
. ---> so $$36$$ = $$2^2$$ and $$3^2$$
Retired Moderator D
Joined: 25 Feb 2013
Posts: 1193
Location: India
GPA: 3.82
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

1
dave13 wrote:
Quote:

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.

Hope it helps.

Hello niks18, could you please advice regarding Bunuels tips about the perfect square. please see below and advice what i missed or did wrong, or counted incorrectly :) many thanks!

lets take 36. here is prime factorization of 36 (2*18) ---> (18) ---> (2*9) ---> (9) --->(3*3)

$$Tips about the perfect square$$

1. The number of distinct factors of a perfect square is ALWAYS ODD. lets take 36 in all cases to test your tips:) - it has 1, 2, 3, 9 = 4 as distinct factors, no but it is EVEN :?

2. The sum of distinct factors of a perfect square is ALWAYS ODD. 36 - so it has 1, 2, 3, 9 as distinct factors hence 1+ 2+3+9 = 15 is it correct ?

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. let me break down this sentence into two clauses:)

A perfect square ALWAYS has an ODD number of Odd-factors ---> odd numbers are 1, 3, 9 so total 3 odd numbers

A perfect square ALWAYS has an EVEN number of Even-factors ---> 2, 8 so in total 2 numbers is corrrect :? but how about 6 ! ?? or is my prime factorization incorrect ??

4. Perfect square always has [b]even powers of its prime factors
. ---> so $$36$$ = $$2^2$$ and $$3^2$$

Hi dave13

you have factorized 36 incorrectly.

$$36=2^2*3^2$$. Hence number of factors $$= (2+1)*(2+1)=9=Odd$$

Factors of 36 are 1,2,3,4,6,9,12,18 & 36. Now figure out the validity of tips yourself.
Non-Human User Joined: 09 Sep 2013
Posts: 12034
Re: How many factors does 36^2 have?  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

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.
_________________ Re: How many factors does 36^2 have?   [#permalink] 07 Feb 2019, 08:55
Display posts from previous: Sort by

# How many factors does 36^2 have?

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

#### MBA Resources  