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Q represents the number of even integers, x, such that 2^20<x<2^30.

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Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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New post 06 Jun 2017, 10:33
1
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A
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E

Difficulty:

  85% (hard)

Question Stats:

47% (02:01) correct 53% (02:19) wrong based on 168 sessions

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Q represents the number of even integers, x, such that \(2^{20}<x<2^{30}\). Which of the following must be true?

I. Q is even.

II. Q is divisible by 31

III. Q is divisible by 32

A. I only
B. II only
C. I and II only
D. I, II and III
E. None of the above

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Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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New post 06 Jun 2017, 12:02
1
The first number after\(2^{20}\) to be divisible by 2 = \(2^{20}\) + 2

The last number before\(2^{30}\) to be divisible by 2 = \(2^{30}\) - 2

Now calculate the total number of number between \(2^{20}\) + 2 and \(2^{30}\) - 2, inclusive

\(2^{20}\) + 2 + (Q-1)2 = \(2^{30}\) - 2

(Q-1)2 = \(2^{30}\) - 2 - \(2^{20}\) - 2

(Q-1) = \(2^{29}\) - 1 - \(2^{19}\) - 1

Q = \(2^{29}\) - 1 - \(2^{19}\)

Q = \(2^{29}\) - \(2^{19}\) - 1

Q = \(2^{19}\) [\(2^{10}\) - 1] - 1

Q = \(2^{19}\)[(\(2^{5}\)+1)(\(2^{5}\)-1)] - 1

Q = \(2^{19}\)[33*31] - 1

Lets see the options
I. Q is even.
Now since Q is and even number - odd, therefore Q has to be odd

II. Q is divisible by 31
As we see, Q is not divisible by 31

III. Q is divisible by 32
As Q is odd, it will not be divisible by 32

Answer is E. None of the above
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Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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New post 19 Jun 2017, 06:02
Hi Quantumlinear,

How can you write (2^20) + 2 + (Q-1)2 = (2^30) - 2 when calculating the number of even numbers between (2^20)+2 and (2^30)-2.?
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Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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New post 19 Jun 2017, 06:53
Hello Sarugiri,

That comes from the formula of AP (Arithmetic Progression).

In an arithmetic progression, the difference between 2 consecutive terms remains constant.

Let the first term be A and the common difference be D
Then
First term = A
Second Term = A+D = A+(2-1)D
Third Term = (A+D)+D=A+2D = A+(3-1)D
...
Nth term = A + (N-1)D = first term + (N-1)*D

Here the nth term is 2^20-2 and first term is 2^20+2 and the common difference is 2.

So we can write 2^20-2=2^20+2 + (N-1)*2

Hope that answers your question
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Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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Re: Q represents the number of even integers, x, such that 2^20<x<2^30. &nbs [#permalink] 23 Jul 2018, 05:05
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