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How many factors of 72^2 are multiples of 6?

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Intern
Joined: 27 Jul 2018
Posts: 5
How many factors of 72^2 are multiples of 6?  [#permalink]

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24 Dec 2018, 01:54
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Difficulty:

55% (hard)

Question Stats:

52% (01:46) correct 48% (01:20) wrong based on 40 sessions

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How many factors of 72^2 are multiples of 6?

A. 35
B. 25
C. 30
D. 24
E. 12
Math Expert
Joined: 02 Aug 2009
Posts: 7737
Re: How many factors of 72^2 are multiples of 6?  [#permalink]

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24 Dec 2018, 04:53
1
07tiger wrote:
How many factors of 72^2 are multiples of 6?

A. 35
B. 25
C. 30
D. 24
E. 12

Let us factorization 72^2..
$$72^2=(2^33^2)^2=2^63^4$$
Number of factors = (6+1)(4+1)=7*5=35
Now 6 is 2*3, so just multiple of just 2 and 3 are (6+1) and (4+1), so total 7+5=12..
But in each case, we have added factor 1 twice, once with 2 and once with 3..
So 12-1=11...
Number of factors that are multiples of 6 are 35-11=24..

D
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Intern
Joined: 27 Jul 2018
Posts: 5
Re: How many factors of 72^2 are multiples of 6?  [#permalink]

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24 Dec 2018, 17:20
Thank you. I was struggling to wrap my head around this one.
Intern
Joined: 13 Mar 2018
Posts: 14
Re: How many factors of 72^2 are multiples of 6?  [#permalink]

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25 Dec 2018, 04:12
chetan2u wrote:
07tiger wrote:
How many factors of 72^2 are multiples of 6?

A. 35
B. 25
C. 30
D. 24
E. 12

Let us factorization 72^2..
$$72^2=(2^33^2)^2=2^63^4$$
Number of factors = (6+1)(4+1)=7*5=35
Now 6 is 2*3, so just multiple of just 2 and 3 are (6+1) and (4+1), so total 7+5=12..
But in each case, we have added factor 1 twice, once with 2 and once with 3..
So 12-1=11...
Number of factors that are multiples of 6 are 35-11=24..

D

Hi chetan 2u

kindly explain this concept
"But in each case, we have added factor 1 twice, once with 2 and once with 3..So 12-1=11..." I am not able to understand
Intern
Joined: 19 Jul 2017
Posts: 7
Re: How many factors of 72^2 are multiples of 6?  [#permalink]

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25 Dec 2018, 09:08
Hi Bunuel,
Could you please explain the concept and solution for this question.
Math Expert
Joined: 02 Aug 2009
Posts: 7737
How many factors of 72^2 are multiples of 6?  [#permalink]

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25 Dec 2018, 09:19
SWAPNILP wrote:
chetan2u wrote:
07tiger wrote:
How many factors of 72^2 are multiples of 6?

A. 35
B. 25
C. 30
D. 24
E. 12

Let us factorization 72^2..
$$72^2=(2^33^2)^2=2^63^4$$
Number of factors = (6+1)(4+1)=7*5=35
Now 6 is 2*3, so just multiple of just 2 and 3 are (6+1) and (4+1), so total 7+5=12..
But in each case, we have added factor 1 twice, once with 2 and once with 3..
So 12-1=11...
Number of factors that are multiples of 6 are 35-11=24..

D

Hi chetan 2u

kindly explain this concept
"But in each case, we have added factor 1 twice, once with 2 and once with 3..So 12-1=11..." I am not able to understand

Hi
what are the factors of 2^6..
Total factors are (6+1) and they are $$1,2,2^2,2^3,2^4,2^5,2^6$$
What about factors of 3^4...they are (4+1) and they are $$1,3,3^2,3^3,3^4$$
So total are 12 but 1 has got repeated twice so cancel out one of them, hence 12-1=11

Hope it helps you.
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How many factors of 72^2 are multiples of 6?   [#permalink] 25 Dec 2018, 09:19
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