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# If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by

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If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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Updated on: 05 May 2013, 22:56
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85% (hard)

Question Stats:

50% (02:22) correct 50% (02:32) wrong based on 197 sessions

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If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by 50 but NOT by 100?

A. 240
B. 345
C. 270
D. 120
E. None of these

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Originally posted by GMATtracted on 05 May 2013, 11:38.
Last edited by Bunuel on 05 May 2013, 22:56, edited 1 time in total.
RENAMED THE TOPIC.
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Re: If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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06 May 2013, 10:12
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GMATtracted wrote:
If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by 50 but NOT by 100?

A. 240
B. 345
C. 270
D. 120
E. None of these

To be divisible by 50, the factor must have $$2*5^2$$ and to be not divisible by 100, it must NOT have $$2^2*5^2$$. Hence the only constraints are on the power of 2 (which must be 1) and the power of 5 (which must be greater than or equal to 2)

The other prime factors can appear in any way in the factor. So number of factors = 1*(6)*(5)*(9) = 270
1 - because 2 can have a power in only one way
6 - because 3 can have 6 different powers (0/1/2/3/4/5)
5 - because 5 can have 5 different powers (2/3/4/5/6)
9 - because 7 can have 9 different powers (0/1/2/3/4/5/6/7/8)

If you are still wondering how we got this product, check out this post: http://www.veritasprep.com/blog/2010/12 ... ly-number/
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05 May 2013, 12:58
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Numbers divisible by 50 not divisible by 100 would be divisible by $$2^1$$ and $$5^2$$, but power of 2 cannot be more than 1 as anything including $$2^2$$ and $$5^2$$ would be divisible by 100. So only 1 power of 2 allowed(2^1) and 5 powers of 5($$5^2$$,$$5^3$$,$$5^4$$,$$5^5$$,$$5^6$$)
=1*6*5*9
(1 powers of 2)*(6 powers of 3)*(5 powers of 5)*(9 powers of 7)
=270
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05 May 2013, 12:59
Another method can be
factors of 50 - factors of 100
1890-1620=270
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Re: If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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21 May 2013, 22:42
VeritasPrepKarishma wrote:
GMATtracted wrote:
If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by 50 but NOT by 100?

A. 240
B. 345
C. 270
D. 120
E. None of these

To be divisible by 50, the factor must have $$2*5^2$$ and to be not divisible by 100, it must NOT have $$2^2*5^2$$. Hence the only constraints are on the power of 2 (which must be 1) and the power of 5 (which must be greater than or equal to 2)

The other prime factors can appear in any way in the factor. So number of factors = 1*(6)*(5)*(9) = 270
1 - because 2 can have a power in only one way
6 - because 3 can have 6 different powers (0/1/2/3/4/5)
5 - because 5 can have 5 different powers (2/3/4/5/6)
9 - because 7 can have 9 different powers (0/1/2/3/4/5/6/7/8)

If you are still wondering how we got this product, check out this post: http://www.veritasprep.com/blog/2010/12 ... ly-number/

Why can't 5 have 6 different powers. Because, you add 1 to the powers of 3 and 7, which I understand. Also, one 2 is understandable because more than one 2 will lead to divisibility by 100.
Thanks!
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Re: If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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21 May 2013, 23:50
sharmila79 wrote:
VeritasPrepKarishma wrote:
GMATtracted wrote:
If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by 50 but NOT by 100?

A. 240
B. 345
C. 270
D. 120
E. None of these

To be divisible by 50, the factor must have $$2*5^2$$ and to be not divisible by 100, it must NOT have $$2^2*5^2$$. Hence the only constraints are on the power of 2 (which must be 1) and the power of 5 (which must be greater than or equal to 2)

The other prime factors can appear in any way in the factor. So number of factors = 1*(6)*(5)*(9) = 270
1 - because 2 can have a power in only one way
6 - because 3 can have 6 different powers (0/1/2/3/4/5)
5 - because 5 can have 5 different powers (2/3/4/5/6)
9 - because 7 can have 9 different powers (0/1/2/3/4/5/6/7/8)

If you are still wondering how we got this product, check out this post: http://www.veritasprep.com/blog/2010/12 ... ly-number/

Why can't 5 have 6 different powers. Because, you add 1 to the powers of 3 and 7, which I understand. Also, one 2 is understandable because more than one 2 will lead to divisibility by 100.
Thanks!

$$50 = 2*5^2$$

The power of 5 must be at least 2. It can be 3, 4, 5 or 6 as well. Here there are 5 different ways in which you can give a power to 5.
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Karishma
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Posts: 58396
Re: If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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22 May 2013, 02:25
GMATtracted wrote:
If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by 50 but NOT by 100?

A. 240
B. 345
C. 270
D. 120
E. None of these

Similar question to practice: how-many-integers-from-1-to-200-inclusive-are-divisib-109333.html

Hope it helps.
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If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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31 Aug 2015, 09:11
here, we have to catch all factors for 50.

100=2^2*5^2
50=2*5^2

Now, to have number divisible by 50 and not by 100 we have to make sure that 2 is taken only once. As, if we pick 2 two's then it will form 100 with no. of 5 selected and since min 5 we have to select is 2, we cannot have 2 more than one.

Number= 2^7*3^5*5^6*7^8

So, there are 6 ways of picking 3, 9 ways of picking 7, 5 ways of picking 5 and only one way of picking 2 and that is 2^1.

Why only 5 ways for picking 5 when we have 7 ways(0,1,2,3,4,5,6)?

Reason: we need 50 so we have to pick 5^2 and greater than it. Also, we can pick only 2^1 because picking 2^2 will form 100 with 5 already selected. For example, 5^2*2^2=100. So, we cannot afford it.

Hence, total number of factors.

= 6(3's)*9(7's)*5(5's)*1(2's)=270

Thanks,
Gaurav
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Re: If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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06 Apr 2016, 11:12
1
If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by 50 but NOT by 100?

50=2*5^2
100=2^2*5^2
N=(2*2^6)*(5^2*5^4)*(3^5)*(7^8)
=(2*5^2)*(2^6*5^4*3^5*7^8)
We dont need any other power of 2 except (2*5^2)
no of factors (power of 5+1)*(power of 3+1)*(power of 7+1)= 5*6*9=270

Hence C
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Re: If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by  [#permalink]

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04 Sep 2019, 04:04
GMATtracted wrote:
If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by 50 but NOT by 100?

A. 240
B. 345
C. 270
D. 120
E. None of these

$$N=2^7*3^5*5^6*7^8$$

num factors of N/50: $$N=(2*5^2)[2^6*3^5*5^4*7^8]=[6+1][5+1][4+1][8+1]=1890$$
num factors of N/100: $$N=(2^2*5^2)[2^5*3^5*5^4*7^8]=[5+1][5+1][4+1][8+1]=1620$$

num factors of N/50 but not N/100: $$1890-1620=270$$

Re: If N=2^7*3^5*5^6*7^8. How many factors of N are divisible by   [#permalink] 04 Sep 2019, 04:04
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