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The number 100^2 has how many factors?

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SW4
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The number 100^2 has how many factors? [#permalink] New post   02 Dec 2016, 14:23
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The number 100^2 has how many factors?

A) 28
B) 25
C) 24
D) 20
E) 16
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B
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BrentGMATPrepNow
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Re: The number 100^2 has how many factors? [#permalink] New post   02 Dec 2016, 14:54
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SW4 wrote:
The number 100^2 has how many factors?

A) 28
B) 25
C) 24
D) 20
E) 16


If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) = (5)(4)(2) = 40

Now onto the question.....
100^2 = [(2)(2)(5)(5)]^2
= [(2)(2)(5)(5)][(2)(2)(5)(5)]
= (2^4)(5^4)
So, the number of positive divisors = (4+1)(4+1) =(5)(5) = 25

Answer: B

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vitaliyGMAT
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Re: The number 100^2 has how many factors? [#permalink] New post   02 Dec 2016, 23:21
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\(100^2 = (2^2*5^2)^2 = 2^4*5^4\)

\(N=p^a*q^b*r^c...\)

Total number of positive factors of \(N = (a+1)(b+1)(c+1)...\)

Hence: \((4+1)(4+1) = 25\)

Answer B
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stonecold
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Re: The number 100^2 has how many factors? [#permalink] New post   28 Dec 2016, 08:30
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Another Approach to this Question ->

100^2 is a perfect Square => A PERFECT SQUARE ALWAYS HAS ODD NUMBER OF FACTORS.

Notice 25 is the only odd number among choices.

Hence B
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Re: The number 100^2 has how many factors? [#permalink] New post   28 Dec 2016, 09:01
SW4 wrote:
The number 100^2 has how many factors?

A) 28
B) 25
C) 24
D) 20
E) 16


\(100^2 = (2^2*5^2)^2 = 2^4*5^4\)

So, No of factors will be \(( 4+1 )( 4+1 ) = 25\)

Hence, the correct answer will be (B) 25
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Re: The number 100^2 has how many factors? [#permalink] New post   04 Apr 2017, 00:11
100= 10*10=2*5*2*5=2^2*5^2

100^2= 2^4*5^4

Total no. of factors= (4+1)(4+1)= 25
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vijaykhot
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Re: The number 100^2 has how many factors? [#permalink] New post   20 Jan 2021, 19:43
Is there any way to calculate the number of factors using the combination formula on the prime factors of the given number??
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Re: The number 100^2 has how many factors? [#permalink]
20 Jan 2021, 19:43
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