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

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

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New post 20 Oct 2017, 08:07
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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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How many factors of 2^3*3^4*5^5 are Even numbers? [#permalink]

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New post 20 Oct 2017, 08:12
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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How many factors of 2^3*3^4*5^5 are Even numbers? [#permalink]

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New post 20 Oct 2017, 09:19
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
\(3^4*5^5\)
Take exponents and add 1
(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

Answer C

*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, 09:19
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