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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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SVP
Joined: 20 Jul 2017
Posts: 1505
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of  [#permalink]

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09 Aug 2019, 06:53
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Difficulty:

55% (hard)

Question Stats:

56% (02:25) correct 44% (02:17) wrong based on 25 sessions

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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
Senior Manager
Joined: 09 Jan 2017
Posts: 314
Location: India
Re: How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of  [#permalink]

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09 Aug 2019, 08:48
6
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

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Intern
Joined: 11 Jun 2019
Posts: 34
Location: India
How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of  [#permalink]

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09 Aug 2019, 08:42
1
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)
How many factors of the number 2^9 * 3^6 * 5^5 * 10^4 are multiples of   [#permalink] 09 Aug 2019, 08:42