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Math Expert V
Joined: 02 Sep 2009
Posts: 58464
Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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32 00:00

Difficulty:   85% (hard)

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

_________________
Math Expert V
Joined: 02 Aug 2009
Posts: 8023
Re: Which of the following must be true?  [#permalink]

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3
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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General Discussion
Senior Manager  G
Joined: 24 Apr 2016
Posts: 321
Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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2
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
Intern  B
Joined: 22 Oct 2016
Posts: 21
Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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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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Regards,
Sarugiri
Intern  S
Joined: 11 Aug 2016
Posts: 46
Location: India
Concentration: Operations, General Management
Schools: HBS '18, ISB '17, IIMA
GMAT 1: 710 Q49 V38 GPA: 3.95
WE: Design (Manufacturing)
Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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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

Intern  B
Joined: 04 May 2018
Posts: 32
Location: India
GMAT 1: 650 Q46 V34 GMAT 2: 680 Q49 V34 GPA: 3.3
Which of the following must be true?  [#permalink]

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 58464
Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

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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

________________________________
Merging topics.
_________________ 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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