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# How many factors of 2^3*3^4*5^5 are Even numbers?

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CR Forum Moderator
Status: The best is yet to come.....
Joined: 10 Mar 2013
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How many factors of 2^3*3^4*5^5 are Even numbers? [#permalink]

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20 Oct 2017, 07:07
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How many factors of $$2^3*3^4*5^5$$ are Even numbers?

A. 20
B. 30
C. 90
D. 100
E. 120
[Reveal] Spoiler: OA

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Hasan Mahmud

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Joined: 25 Feb 2013
Posts: 922
Location: India
GPA: 3.82
How many factors of 2^3*3^4*5^5 are Even numbers? [#permalink]

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20 Oct 2017, 07:12
Mahmud6 wrote:
How many factors of $$2^3*3^4*5^5$$ are Even numbers?

A. 20
B. 30
C. 90
D. 100
E. 120

Total number of factors $$= 4*5*6=120$$

Total number of Odd factors i.e $$3^4*5^5$$ is $$5*6=30$$

Hence number of even factors $$= 120-30=90$$

Option C
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Joined: 22 May 2016
Posts: 1331
How many factors of 2^3*3^4*5^5 are Even numbers? [#permalink]

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20 Oct 2017, 08:19
Mahmud6 wrote:
How many factors of $$2^3*3^4*5^5$$ are Even numbers?

A. 20
B. 30
C. 90
D. 100
E. 120

Method:
(# of even factors) =
(Total # of factors) - (# of odd factors)

1) Call the number "Z."
Use the prime factorization
$$2^3*3^4*5^5$$

2) Add 1 to each prime's exponent
(3+1) = 4
(4+1) = 5
(5+1) = 6

3) Multiply the results
4 * 5 * 6 = 120
Z has 120 factors, including Z and 1

4) Number of odd factors? Remove 2; it's even. Same method
$$3^4*5^5$$
(4 + 1) = 5
(5 + 1) = 6
Multiply the results.
5 * 6 = 30 odd factors

5) Number of even factors
(Total) - (odd) = even
120 - 30 = 90 even factors

*The overall theory is here, scroll to "Finding the Number of Factors of an Integer."
For number of even (or odd) factors, see Bunuel 's explanation in this problem

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How many factors of 2^3*3^4*5^5 are Even numbers?   [#permalink] 20 Oct 2017, 08:19
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