If x is a positive integer, what is the remainder when x is divided by

Question

If x is a positive integer, what is the remainder when x is divided by 7?

(1) The remainder when x is divided by 4 is 3
(2) The remainder when x is divided by 5 is 1

BrentGMATPrepNow · Accepted Answer

Target question :    What is the remainder when x is divided by 7?

Statement 1 : The remainder when x is divided by 4 is 3  
When it comes to remainders, we have a nice rule that says: 
  If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.    
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

So, if the remainder when x is divided by 4 is 3, then some possible values of x are:  3, 7, 11, 15, 19, 23,  etc
Let's examine some of these possible values of x
 Case a : If x =  3 , then  the remainder when x is divided by 7 is  3  
 Case b : If x =  7 , then  the remainder when x is divided by 7 is  0  
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : The remainder when x is divided by 5 is 1  
Some possible values of x are:  1, 6, 11, 16, 21, 26, 31,  etc
Let's examine some of these possible values of x
 Case a : If x =  1 , then  the remainder when x is divided by 7 is  1  
 Case b : If x =  6 , then  the remainder when x is divided by 7 is  6  
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Statement 1 tells us that some possible values of x are:  3, 7,  11 , 15, 19, 23, 27,  31 , 35, 39,  etc
Statement 2 tells us that some possible values of x are:  1, 6,  11 , 16, 21, 26,  31 ,  etc
Since we already have 2 possible x-values that BOTH statements share, let's examine the following cases:

Case a : If x =   11  , then  the remainder when x is divided by 7 is  4  
 Case b : If x =   31  , then  the remainder when x is divided by 7 is  3  
Since we still cannot answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer:  E

RELATED VIDEO FROM OUR COURSE
 https://www.youtube.com/watch?v=XCxgIR042tE

niks18 · Answer

Statement 1: x=4k+3, where k is any integer. Thus there could be multiple value of x and hence multiple remainders when divided by 7. Insufficient

Statement 2: x=5q+1, where q is any integer. Thus there could be multiple value of x and hence multiple remainders when divided by 7. Insufficient

Combining 1 and 2 we know x is of the form 4k+3 & 5q+1, so x can be 11 or 51, for k=q=2 and k=12 & q=10. so again multiple remainders possible. Hence Insufficient

Option  E

JeffTargetTestPrep · Answer

We need to determine the remainder of x/7.

Statement One Alone:

The remainder when x is divided by 4 is 3.

We see that x can be values such as 3, 7, 11, 15, 19, 23, 27, 31, etc.

However, using those values, we get various remainders when dividing by 7. For example, when 3 is divided by 7, the remainder is 3, but when 7 is divided by 7, the remainder is 0. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

The remainder when x is divided by 5 is 1.

We see that x can be values such as 1, 6, 11, 16, 21, 26, 31, etc.

However, using those values, we get various remainders when dividing by 7. For example, when 1 is divided by 7, the remainder is 1, but when 6 is divided by 7, the remainder is 6. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the two statements together, we see that x could be 11 or 31. When x = 11, the remainder is 4 when 11 is divided by 7. However, when x = 31, the remainder is 3 when 31 is divided by 7.

Answer: E

adkikani · Answer

niks18   gmatbusters   Bunuel   chetan2u   VeritasPrepKarishma   amanvermagmat

Is there any other approach than plugging numbers to solve this Q?

GMATBusters · Answer

Hi
The approach taken by GMATprepnow is a nice one.
Try to practice and be used to this approach, this will help in similar questions.

adkikani · Answer

gmatbusters

I could not understand approach by  GMATPrepNow  , esp this quote:

I understood  niks18  approached this by using:

Dividend = (Integer Quotient) * Divisor + Remainder

Let me know if my understanding is correct or flawed.

GMATBusters · Answer

Hi
Both are same approach, it is the matter of presentation that differs.
"If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc. "
Example to explain this: Here D is Divisor, R is Remainder
let N when divided by 2 gives remainder as 1.
then N can be 1, 2+1, 2*2+1, 3*2+1...
N can be 1, 3, 5, 7... so on...

Grasp the concept, terminology doesn't matter much.

BrentGMATPrepNow · Answer

Here are three important things you need to know about remainders:

If N divided by D equals Q with remainder R, then N = DQ + R  
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.    
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D  
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

RELATED VIDEO
 https://www.youtube.com/watch?v=GFXrkH9UMGM

Cheers, 
Brent

KarishmaB · Answer

Here, 
Stmnt 1: x = 4a + 3
So x = 3, 7, 11, 15...
Stmnt 2: x = 5b + 1
So x = 1, 6, 11, 16 ...

First number of both types is 11. Next such number will be 11 + 4*5 = 31.
11 divided by 7 will give remainder 4. 
31 divided by 7 will give remainder 3 and so on...

So answer is (E)

Vordhosbn · Answer

Hey  JeffTargetTestPrep ,  BrentGMATPrepNow

Is there a better approach than listing numbers all the way to 31? I feel like there must be a better approach than this. On the actual exam, under time pressure, one might not unroll values all the way to 31.

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.

If x is a positive integer, what is the remainder when x is divided by

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 Data Sufficiency (DS) Questions

If x is a positive integer, what is the remainder when x is divided by

Prep Toolkit

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