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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of

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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of  [#permalink]

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New post 05 Dec 2019, 04:20
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D
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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of  [#permalink]

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New post 05 Dec 2019, 04:30
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How many factors of the number \(2^{6}∗3^{5}∗5^{4}∗6^{3}\) are multiples of 360?

\(360= 2^{3}*3^{2}*5\)

--> \(\frac{(2^{6}∗3^{5}∗5^{4}∗6^{3})}{(2^{3}*3^{2}*5)}= 2^{6}*3^{6}*5^{3}\)

The number of factors (6+1)(6+1)(3+1)= 49*4= 196

The answer is D
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Re: How many factors of the number 2^6*3^5*5^4*6^3 are multiples of  [#permalink]

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New post 05 Dec 2019, 04:35
Prime Factors of 2^6∗3^5∗5^4∗6^3 = 2^9∗3^8∗5^4
Prime Factors of 360 = 3*2*3*2*2*5

now 360 is the multiple 2^6∗3^5∗5^4∗6^3
or 2^6∗3^5∗5^4∗6^3 = 360M
M= 2^6∗3^6∗5^3

No of Factors of M = (6+1)(6+1)(3+1) = 7* 7 *4 = 196

Answer: D
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Re: How many factors of the number 2^6*3^5*5^4*6^3 are multiples of  [#permalink]

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New post 05 Dec 2019, 04:43
Bunuel wrote:
How many factors of the number \(2^6*3^5*5^4*6^3\) are multiples of 360?

A. 36
B. 108
C. 144
D. 196
E. 288


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Any multiple of 360 can be written as 360*k, where k is an integer.

On prime factorizing 360, we get = \(2^3 *3^2 * 5^1\)

So, we can write the given number \(2^6*3^5*5^4*6^3\) as : \((2^3 * 3^2 * 5^1) * (2^6 * 3^6 * 5^3) = 360* (2^6 * 3^6 * 5^3)\)

Now, we just need to find the total number of factors of \((2^6 * 3^6 * 5^3)\). Because all the factors when multiplied with 360 will all be multiples of 360. :)

Total factors of \([2^6 * 3^6 * 5^3]\)

\(= (6 + 1) (6 + 1) (3 + 1)\)

\(= 7 * 7 * 4\)

\(= 196\)

Correct answer is Option D.
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Re: How many factors of the number 2^6*3^5*5^4*6^3 are multiples of  [#permalink]

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New post 05 Dec 2019, 09:00
360= 2^3*3^2*5
to determine factors of \(2^6*3^5*5^4*6^3\) which are multiple of 360 ;
\(2^6*3^5*5^4*6^3\) /2^3*3^2*5 = 2^6*3^6*5^3 ; 7*7*4 ; 196
IMO D

Bunuel wrote:
How many factors of the number \(2^6*3^5*5^4*6^3\) are multiples of 360?

A. 36
B. 108
C. 144
D. 196
E. 288


Are You Up For the Challenge: 700 Level Questions
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Re: How many factors of the number 2^6*3^5*5^4*6^3 are multiples of  [#permalink]

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New post 08 Dec 2019, 20:48
Bunuel wrote:
How many factors of the number \(2^6*3^5*5^4*6^3\) are multiples of 360?

A. 36
B. 108
C. 144
D. 196
E. 288


Are You Up For the Challenge: 700 Level Questions


Breaking the given factors down further, we have:

2^6 x 3^5 x 5^4 x 2^3 x 3^3

2^9 x 3^8 x 5^4

and

360 = 36 x 10 = 6 x 6 x 2 x 5 = 2^3 x 3^2 x 5^1

Since (2^9 x 3^8 x 5^4)/(2^3 x 3^2 x 5^1) = 2^6 x 3^6 x 5^3, there are (6 + 1) (6 + 1) (3 + 1) = 7 x 7 x 4 = 196 multiples of 360.

Answer: D
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Re: How many factors of the number 2^6*3^5*5^4*6^3 are multiples of   [#permalink] 08 Dec 2019, 20:48
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