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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of 360?

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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of 360? [#permalink]

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New post 20 Oct 2017, 07:35
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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
[Reveal] Spoiler: OA

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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of 360? [#permalink]

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Mahmud6 wrote:
How many factors of the number \(2^6*3^5*5^4*6^3\) are multiples of 360?

A. 36
B. 1088
C. 144
D. 196
E. 288


\(2^6*3^5*5^4*6^3=2^9*3^8*5^4\)

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

all factors of \(2^9*3^8*5^4\) that can be written as multiples of \(360\) will be of the form \(2^3*3^2*5*p\)

therefore \(2^9*3^8*5^4=2^3*3^2*5*p\)

or \(p= 2^6*3^6*5^3\)

Now the number of factors of \(p=(1+6)*(1+6)*(1+3)=196\)

Option D

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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of 360? [#permalink]

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New post 20 Oct 2017, 07:47
To find out the number of multiples of 360, let's first find out its factors and then take them out of the total factors. This will ensure that the remaining factors will always include the factors of 360.

Let's solve now:

360 = 9 * 4 * 10 = \(3^2*2^3*5^1\)

That means let's take two 3s, three 2s and one 5 out of the original number.

\(2^6*3^5*5^4*6^3\)

After taking the required factors out, we are left with : \(2^3*3^3*5^3*6^3\)

As per the formula, \(a^p*b^q*c^r\), we have number of factors = (p+1)(q+1)(r+1), where a,b and c MUST be distinct primes.

Let's prime factorise all and club the common number, I will have the number as \(2^6*3^6*5^4\) [Confused? Just split take 6 = 2*3 and then join 2 with existing 2s and 3 with existing 3s]

Therefore, for our question I can say, number of factors multiple of 360 = (6+1) ( 6+1)(3+1) = 49*4 = 196

Hence, answer must be D.
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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of 360?   [#permalink] 20 Oct 2017, 07:47
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How many factors of the number 2^6*3^5*5^4*6^3 are multiples of 360?

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