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How many factors does 36^2 have?

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Intern
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Joined: 24 Jul 2010
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How many factors does 36^2 have?  [#permalink]

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New post 15 Aug 2010, 02:47
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How many factors does 36^2 have?

A. 2
B. 8
C. 24
D. 25
E. 26
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Re: Help: Factors problem !!  [#permalink]

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New post 15 Aug 2010, 03:04
3
4
praveengmat wrote:
How many factors does 36^2 have?
A 2
B 8
C 24
D 25
E 26
Please help as to how to solve this problem with 1 minute !!


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.

Back to the original question:

How many factors does 36^2 have?

\(36^2=(2^2*3^2)^2=2^4*3^4\) --> # of factors \((4+1)*(4+1)=25\).

Answer: D.

Or another way: 36^2 is a perfect square, # of factors of perfect square is always odd (as perfect square has even powers of its primes and when adding 1 to each and multiplying them as in above formula you'll get the multiplication of odd numbers which is odd). Only odd answer in answer choices is 25.

Hope it helps.
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Re: Help: Factors problem !!  [#permalink]

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New post 15 Aug 2010, 03:11
Bunuel wrote:
praveengmat wrote:
How many factors does 36^2 have?
A 2
B 8
C 24
D 25
E 26
Please help as to how to solve this problem with 1 minute !!


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.

Back to the original question:

How many factors does 36^2 have?

\(36^2=(2^2*3^2)^2=2^4*3^4\) --> # of factors \((4+1)*(4+1)=25\).

Answer: D.

Or another way: 36^2 is a perfect square, # of factors of perfect square is always odd (as perfect square has even powers of its primes and when adding 1 to each and multiplying them as in above formula you'll get the multiplication of odd numbers which is odd). Only odd answer in answer choices is 25.

Hope it helps.



Thanks a ton !!.. loved the approach !
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Re: Help: Factors problem !!  [#permalink]

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New post 14 Oct 2010, 13:58
1
Factors of a perfect square can be derived by using prime factorization and then using the formula to find perfect square's factors.

In this case \((36)^2= (2^2*3^2)^2=2^4*3^4\) or \((36)^2=(6^2)^2=(6)^4=(2*3)^4=2^4*3^4\)

And now you can use the formula explained above by Bunuel to determine the answer, which is \((4+1)*(4+1)=5*5=25=Odd\)(Trick is there must be odd number of factors of a perfect square and only 25 is odd in answer choices, so it can be solved within 30 seconds or less :) )

Please! go through the GMAT Math Book by GMAT CLUB (written by bunuel & walker), all of these tips & tricks are written there. (even I have compiled them in one .pdf file and is shared here on Math forum)
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Re: Help: Factors problem !!  [#permalink]

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New post 29 Aug 2011, 09:57
This method is worth bookmarking. Appreciate it
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Re: Help: Factors problem !!  [#permalink]

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New post 01 Sep 2011, 19:17
The Easy Answer! (Applicable only in case of perfect square numbers)
A perfect square always have a odd number of factors.
36^2 is a perfect square.
Given the answer choices, the only odd number of factor is 25.

So, The definite answer is D.
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Re: Help: Factors problem !!  [#permalink]

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New post 02 Sep 2011, 08:08
36^2 = 2^4 3^4

total factors = (4+1)(4+1) = 25.

Answer is D.
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Re: Help: Factors problem !!  [#permalink]

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New post 03 Sep 2011, 10:04
The odd number of factors for perfect squares solves this in no time. Nice trick.
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Re: How many factors does 36^2 have?  [#permalink]

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New post 25 Oct 2014, 10:35
praveengmat wrote:
How many factors does 36^2 have?

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


This can be solved, by just looking at the answer choices. Since the given number is a square, it must have an odd number of factors. Only D has odd.

Ans. D

If we need to solve it, then
\((36)^2 = (2^4*3^4)\)
For \(a^n*b^m\) the number of factors is defined by \(((n+1)(m+1))\)
\(= (4+1)(4+1)\)
\(= 25\)

Ans. D
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How many factors does 36^2 have?  [#permalink]

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New post 24 Aug 2018, 10:12
praveengmat wrote:
How many factors does 36^2 have?

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


OA:D
\(36^2=6^4=2^4*3^4\)
Number of Factors \(= (4+1)*(4+1) = 5*5 =25\)

Furthermore , Square of a number always has odd number of factors.
For ex \(4= 2^2\), Factors : \(1,2,4\)
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Re: How many factors does 36^2 have?  [#permalink]

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New post 24 Aug 2018, 10:15
praveengmat wrote:
How many factors does 36^2 have?

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

\(36 = 3^2*2^2\)

So, \(36^2 = 3^4*2^4\)

No of factors of 36^2 is ( 4 +1 )(4 + 1 ) = 25, Answer must be (D)
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Re: How many factors does 36^2 have? &nbs [#permalink] 24 Aug 2018, 10:15
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