Author 
Message 
TAGS:

Hide Tags

Director
Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 527
Location: United Kingdom
Concentration: International Business, Strategy
GPA: 2.9
WE: Information Technology (Consulting)

How many factors does 36^2 have? [#permalink]
Show Tags
22 Jan 2012, 15:48
1
This post received KUDOS
16
This post was BOOKMARKED
Question Stats:
70% (00:24) correct 30% (00:41) wrong based on 520 sessions
HideShow timer Statistics
How many factors does 36^2 have? A. 2 B. 8 C. 24 D. 25 E. 26
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Best Regards, E.
MGMAT 1 > 530 MGMAT 2> 640 MGMAT 3 > 610 GMAT ==> 730
Last edited by Bunuel on 31 Jul 2012, 01:53, edited 1 time in total.
Edited the question and added the OA.



Math Expert
Joined: 02 Sep 2009
Posts: 43831

Re: Factors of 36^2 [#permalink]
Show Tags
22 Jan 2012, 16:07
20
This post received KUDOS
Expert's post
17
This post was BOOKMARKED
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 IntegerFirst 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: mathnumbertheory88376.htmlBACK 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 Oddfactors, and EVEN number of Evenfactors. 4. Perfect square always has even powers of its prime factors. Hope it helps.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Director
Status: Done with formalities.. and back..
Joined: 15 Sep 2012
Posts: 634
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
29 Nov 2012, 02:05
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: 25
GMAT Date: 01192013

Re: Factors of 36^2 [#permalink]
Show Tags
06 Dec 2012, 12:51
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
Joined: 02 Sep 2009
Posts: 43831

Re: Factors of 36^2 [#permalink]
Show Tags
07 Dec 2012, 01:15



Intern
Joined: 21 Oct 2012
Posts: 25
GMAT Date: 01192013

Re: Factors of 36^2 [#permalink]
Show Tags
07 Dec 2012, 06:15
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
Joined: 02 Sep 2009
Posts: 43831

Re: Factors of 36^2 [#permalink]
Show Tags
07 Dec 2012, 06:24



Current Student
Joined: 15 Jan 2014
Posts: 89
Concentration: Healthcare, Strategy

Re: How many factors does 36^2 have? [#permalink]
Show Tags
01 May 2014, 20:18
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
Joined: 02 Sep 2009
Posts: 43831

Re: How many factors does 36^2 have? [#permalink]
Show Tags
02 May 2014, 00:51
1
This post received KUDOS
Expert's post
2
This post was BOOKMARKED
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: howmanyfactorsdoes362have126422.html#p1032696Tips 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 Oddfactors, and EVEN number of Evenfactors. The reverse is also true: if a number has an ODD number of Oddfactors, and EVEN number of Evenfactors 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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 2181
Location: United States (CA)

Re: How many factors does 36^2 have? [#permalink]
Show Tags
25 Sep 2017, 16:59
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 WoodburyStewart
Founder and CEO
GMAT Quant SelfStudy Course
500+ lessons 3000+ practice problems 800+ HD solutions



Intern
Joined: 19 Jun 2017
Posts: 4

Re: How many factors does 36^2 have? [#permalink]
Show Tags
29 Jan 2018, 12:13
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
Joined: 02 Sep 2009
Posts: 43831

Re: How many factors does 36^2 have? [#permalink]
Show Tags
29 Jan 2018, 20:19



Intern
Joined: 19 Jun 2017
Posts: 4

Re: How many factors does 36^2 have? [#permalink]
Show Tags
30 Jan 2018, 02:58
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.



Manager
Joined: 09 Mar 2016
Posts: 227

How many factors does 36^2 have? [#permalink]
Show Tags
30 Jan 2018, 10:14
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 Oddfactors, and EVEN number of Evenfactors. 4. Perfect square always has even powers of its prime factors.
Hope it helps.
Hello niks18, could you please advice regarding Bunuel`s 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 Oddfactors, and EVEN number of Evenfactors. let me break down this sentence into two clauses:)
A perfect square ALWAYS has an ODD number of Oddfactors > odd numbers are 1, 3, 9 so total 3 odd numbers
A perfect square ALWAYS has an EVEN number of Evenfactors > 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 \(3^2\)



PS Forum Moderator
Joined: 25 Feb 2013
Posts: 928
Location: India
GPA: 3.82

Re: How many factors does 36^2 have? [#permalink]
Show Tags
30 Jan 2018, 10:36
1
This post received KUDOS
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 Oddfactors, and EVEN number of Evenfactors. 4. Perfect square always has even powers of its prime factors.
Hope it helps.
Hello niks18, could you please advice regarding Bunuel`s 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 Oddfactors, and EVEN number of Evenfactors. let me break down this sentence into two clauses:)
A perfect square ALWAYS has an ODD number of Oddfactors > odd numbers are 1, 3, 9 so total 3 odd numbers
A perfect square ALWAYS has an EVEN number of Evenfactors > 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 dave13you 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.




Re: How many factors does 36^2 have?
[#permalink]
30 Jan 2018, 10:36






