GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 20 Feb 2019, 19:44

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT Prep Hour

February 20, 2019

February 20, 2019

08:00 PM EST

09:00 PM EST

Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST

February 21, 2019

February 21, 2019

10:00 PM PST

11:00 PM PST

Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

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

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 53020
Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

### Show Tags

06 Jun 2017, 10:33
1
26
00:00

Difficulty:

85% (hard)

Question Stats:

45% (01:58) correct 55% (02:12) wrong based on 226 sessions

### HideShow timer Statistics

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
Joined: 02 Aug 2009
Posts: 7334
Re: Which of the following must be true?  [#permalink]

### Show Tags

30 Dec 2018, 22:24
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
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html
4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html

GMAT Expert

##### General Discussion
Senior Manager
Joined: 24 Apr 2016
Posts: 331
Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

### Show Tags

06 Jun 2017, 12:02
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
Joined: 22 Oct 2016
Posts: 22
Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

### Show Tags

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.?
_________________

Regards,
Sarugiri

Intern
Joined: 11 Aug 2016
Posts: 47
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]

### Show Tags

19 Jun 2017, 06:53
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

Intern
Joined: 04 May 2018
Posts: 28
GMAT 1: 650 Q46 V34
Which of the following must be true?  [#permalink]

### Show Tags

30 Dec 2018, 21:06
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
Math Expert
Joined: 02 Sep 2009
Posts: 53020
Re: Q represents the number of even integers, x, such that 2^20<x<2^30.  [#permalink]

### Show Tags

31 Dec 2018, 00:10
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, 00:10
Display posts from previous: Sort by