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

Question

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

chetan2u · Accepted Answer

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

quantumliner · Answer

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}   - 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

sarugiri · Answer

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.?

tapabrata · Answer

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

Alec0 · Answer

Why subtract by 1 below? Shouldn't we add by 1 to get an inclusive set? The answer is the same either way, but I want to make sure I address a conceptual gap if I have one.

Thanks,
Alec

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions