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How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of

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How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of  [#permalink]

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New post 09 Aug 2019, 07:53
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How many factors of the number \(2^9*3^6*5^5*10^4\) are multiples of 60?

A. 594
B. 648
C. 980
D. 1200
E. 2100
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Re: How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of  [#permalink]

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New post 09 Aug 2019, 09:48
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Dillesh4096 wrote:
How many factors of the number \(2^9*3^6*5^5*10^4\) are multiples of 60?

A. 594
B. 648
C. 980
D. 1200
E. 2100



Lets have 10^4 as 2^4*5^4

so combined we have the number as 2^13*3^6*5^9

prime factorization of 60 we have = 2^2*5*3

so, number of factors of the above number when it is a multiple of 60 is if we calculate the number of factors when we have taken out the exponents of prime factors of 60 from them, basically dividing it!!

now we have - 2^11*3^5*5^8

no. of factors of the above number is (11+1)*(5+1)*(8+1) = 648

Hence, answer is B
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How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of  [#permalink]

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New post 09 Aug 2019, 09:42
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My approach-
2^9 x 3^6 x 5^5 x 10^4 can be written as 2^13 x 3^6 x 5^9.
60= 2^2 x 3^1 x 5^1

Now to find number of multiples of 60 in given number, 2 can have any number of powers between 2 to 13. Hence 2 can be raised to 12 numbers (2,3,4,5,6,7,8,9,10,11,12,13)= 12
Similarly, 3 can be raised to any number between 1 to 6 (1,2,3,4,5,6)=6
and 5 can be raised to any number between 1 to 9 (1,2,3,4,5,6,7,8,9)= 9

Hence total possible numbers are- 12 X 6 X 9 = 648
Hence (B)
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How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of   [#permalink] 09 Aug 2019, 09:42
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How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of

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