How many factors does 36^2 have? : GMAT Problem Solving (PS)
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# How many factors does 36^2 have?

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

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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
[Reveal] Spoiler: OA
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Re: Help: Factors problem !! [#permalink]

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15 Aug 2010, 03:04
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praveengmat wrote:
How many factors does 36^2 have?
A 2
B 8
C 24
D 25
E 26

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 $$(4+1)*(4+1)=25$$.

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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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

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 $$(4+1)*(4+1)=25$$.

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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14 Oct 2010, 13:58
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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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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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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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02 Sep 2011, 08:08
36^2 = 2^4 3^4

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

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Re: Help: Factors problem !! [#permalink]

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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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25 Oct 2014, 05:23
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Re: How many factors does 36^2 have? [#permalink]

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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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Re: How many factors does 36^2 have?   [#permalink] 25 Oct 2014, 10:35
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