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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, 11:33
2
32
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

44% (01:56) correct 56% (02:12) wrong based on 247 sessions

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Re: Which of the following must be true?  [#permalink]

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New post 30 Dec 2018, 23:24
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ajtmatch wrote:
Q represents the number of even integers, x, such that 2^20<x<2^30 then 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


Number of even integers = \(\frac{2^{30}-2^{20}}{2}-1=\frac{2^{20}(2^{10}-1)}{2}-1=2^{19}(2^{10}-1)-1=2^{19}(2^5-1)(2^5+1)-1=2^{19}*31*33-1\)

Now 2^19*31*33 is multiple of 2 and 31, so when we subtract 1 from it, the result will be odd number and also not divisible by 31..

Thus, none of the options are true...

E
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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, 13:02
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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, 07:02
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, 07:53
1
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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Which of the following must be true?  [#permalink]

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New post 30 Dec 2018, 22:06
Q represents the number of even integers, x, such that 2^20<x<2^30 then 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 31 Dec 2018, 01:10
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Re: Q represents the number of even integers, x, such that 2^20<x<2^30.   [#permalink] 31 Dec 2018, 01:10
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Q represents the number of even integers, x, such that 2^20<x<2^30.

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