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The product of all positive even numbers less than or equal to 20 is

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The product of all positive even numbers less than or equal to 20 is  [#permalink]

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New post 13 Dec 2015, 04:09
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A
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D
E

Difficulty:

  95% (hard)

Question Stats:

35% (01:43) correct 65% (01:41) wrong based on 161 sessions

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Re: The product of all positive even numbers less than or equal to 20 is  [#permalink]

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New post 14 Dec 2015, 05:14
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Bunuel wrote:
The product of all positive even numbers less than or equal to 20 is closest to which of the following?

A. 10^6
B. 10^7
C. 10^8
D. 10^9
E. 10^10



2 * 4 * 6 * 8 * 10 * 12 * 14 * 16 * 18 * 20

Replacing closest digits to 10 (6,8,12,14) with 10 and closest to 20 (16,18) with 20.

==>(2 * 4) * (10 * 10 * 10 * 10 * 10) * (20 * 20 * 20)
==>8 * (10^5) * (8 * 10^3)
==>64 * 10^8
==>64 is closest to 50
==>50 * 10^ 8
==>5 * 10^9

Answer D
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The product of all positive even numbers less than or equal to 20 is  [#permalink]

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New post 28 Dec 2015, 14:50
1
\(n = 2*4*6*8*10*12*14*16*18*20
= (2^1^0)(1*2*3*4*5*6*7*8*9*10) = (2^1^0)10!\)

Now:
\(1*10 = 10\)
\(2*5 = 10\)
\(3*4 = 10\) (More than 10)
\(6*8 = 50\) (Less than 50)
\(9*7 = 60\) (More than 60)
\(2^1^0 = 1000\) (More than 1000)

Therefore:
\(n = 1000*(3*10^6) = 3*10^9\), which is closer to \(10^9\)

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Re: The product of all positive even numbers less than or equal to 20 is  [#permalink]

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New post 03 Nov 2018, 16:54
Bunuel wrote:
The product of all positive even numbers less than or equal to 20 is closest to which of the following?

A. 10^6
B. 10^7
C. 10^8
D. 10^9
E. 10^10


2 x 4 x 6 x … x 20 = (2 x 1) x (2 x 2) x (2 x 3) x … x (2 x 10) = 2^10(1 x 2 x 3 x … x 10) = 2^10 x 10!

2^10 = 1024 is approximately 10^3.

10! = 7! x 8 x 9 x 10 = 5040 x 720 is approximately 5000 x 700 = 35 x 10^5. Since 35 is less than 50, we can round it down to 10, so 35 x 10^5 is approximately 10 x 10^5 = 10^6.

Therefore, 2^10 x 10! is approximately 10^3 x 10^6 = 10^9.

Alternate Solution:

We can group the numbers in the product to create 10s or 100s.

2 x 4 x 6 x 8 x 10 x 12 x 14 x 16 x 18 x 20 can be broken down into:

10 = 10^1

8 x 12 = 96 ≈ 100 = 10^2

6 x 16 = 96 ≈ 100 = 10^2

2 x 4 x 14 = 112 ≈ 10^2

The remaining two factors are 18 x 20 = 360 = 3.6 x 10^2

We combine all of these factors, obtaining 3.6 x 10^9, which is closest to answer choice D.

Answer: D
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Re: The product of all positive even numbers less than or equal to 20 is &nbs [#permalink] 03 Nov 2018, 16:54
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