When positive integer x is divided by 5, the remainder is 2.

Question

When positive integer x is divided by 5, the remainder is 2. When positive integer y is divided by 4, the remainder is 1. Which of the following cannot be the sum x + y?

A. 12 
B. 13 
C. 14 
D. 16 
E. 21

ajdse22 · Accepted Answer

One way to solve this problem:-
since x=5a+2, where a is an integer.
and y=4b+1 where b is an integer
x+y=5a+4b+3..

now test for integer solutions of a and b. We need to subtract 3 from answer choice and then test whether the resultant can be expressed 5a+4b.
choice a) 12. Subtract 3 we get 9 . 9 = 4a+5b. take a=1, b=1. 
choice b) 13. Subtract 3 we get 10. 10=4a+5b. take a=0 and b=2. 
choice c) 14. Subtract 3 we get 11. 11 = 4a+5b. if we take a=0, 1,2, can we have b which is positive integer. let us check
a=0 ;11=5b , a = 1 ; 7 = 5b, a = 2 ; 3= 5b, a = 3 , now in this case -1 = 5b.. now from here on we can get negative values on the L.H.S. So we cannot express 11 = 4a+5b where a and b are integers. This is our answer.
choice d) 16 . Subtract 3 we get 13 . 13= 4a+5b. take a =2 and b = 1. 
choice e) 21. subtract 3 we get 18. 18= 4a+5b. take a=2, b=2. Done!

Bunuel · Answer

When positive integer x is divided by 5, the remainder is 2: x could be 2, 7, 12, 17, ... When positive integer y is divided by 4, the remainder is 1: y could be 1, 5, 9, 13, 17, ... If x is 7 and y is 5, then x + y = 12. Eliminate A. If x is 12 and y is 1, then x + y = 13. Eliminate B. If x is 7 and y is 9, then x + y = 16. Eliminate D. If x is 12 and y is 9, then x + y = 21. Eliminate E. By process of elimination the correct answer must be C. Answer: C P.S. Please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to rules 3 and 7. Thank you.

PareshGmat · Answer

Answer = C = 14

Please refer screenshot below for possible combinations

law258 · Answer

y=5r+2 (2,7,12,17,....)

x=4r+1 (1,5,9,13,....)

C is the only number you cannot create using any combination of x and y

bkastan · Answer

Bunuel  how do you know the possible values of X? How does 2/5 give a remainder of 2?

Bunuel · Answer

1.  If  x  and  y  are  positive integers , there exist unique integers  q  and  r , called the  quotient  and  remainder , respectively, such that  y =divisor*quotient+remainder= xq + r  and  0\leq{r}<x .

So, positive integer x is divided by 5, the remainder is 2 means that x = 5q + 2, thus x could be 2, 7, 12, ...

2. Let me ask you a question: how many leftover apples would you have if you had 2 apples and wanted to distribute in 5 baskets evenly? Each basket would get 0 apples and 2 apples would be leftover (remainder).

When a divisor is more than dividend, then the remainder equals to the dividend, for example:
3 divided by 4 yields the reminder of 3:  3=4*0+3 ;
9 divided by 14 yields the reminder of 9:  9=14*0+9 ; 
1 divided by 9 yields the reminder of 1:  1=9*0+1 .

For more on this check:

5. Divisibility/Multiples/Factors 
     Theory   
 Divisibility and Remainders on the GMAT 
 Tips and hints: Divisibility/Multiples/Factors 
 3 Deadly Mistakes you must avoid in LCM-GCD Questions 
 How Well Do You Know Your Factors? 
 THE MOST EFFICIENT WAY TO STUDY LEAST COMMON MULTIPLES ON THE GMAT 
     Questions   
 The E-GMAT Primes Trio - 3 Ques on No. of factors & Prime factors 
 DS Questions 
 PS Questions

6. Remainders 
     Theory   
 REMAINDERS 
 Remainders: Tips and hints 
 Divisibility and Remainders on the GMAT 
 Remainders on the GMAT 
 A Remainders Shortcut for the GMAT 
 All About Negative Remainders on the GMAT 
 A Tricky Question on Negative Remainders 
 Compilation of tips and tricks to deal with remainders 
 Cyclicity and the remainders on the GMAT 
 GMAT Question Patterns: Remainders 
     Questions   
 Units digits, exponents, remainders problems  
 DS Questions 
 PS Questions

For other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

bkastan · Answer

Bunuel  thank you!!!

Posted from my mobile device

gracie · Answer

x=5q+2
y=4p+1
x+y=5q+4p+3
12: q=1;p=1
13: q=2;p=0
16: q=1;p=2
21: q=2;p=2
14: no
C

AlexGmat2017 · Answer

Note that 1 + 2 = 3. Deduct 3 from every proposed answer and try to construct the resulting figure with 5s and 4s.

A. 12 -3 = 9 = 5+4
B. 13 -3 =10= 5+ 5
C. 14 -3 = 11= correct answer
D. 16 -3 = 13 = 5+4+4
E. 21 -3 = 18= 4+4 +5+5

ScottTargetTestPrep · Answer

We can create the following equations:

x = 5Q + 2

Some values for x are 2, 7, 12, 17.

y = 4Z + 1

Some values for y are 1, 5, 9, 13, 17.

Let’s go through each answer choice:

A) 12

Since 7 + 5 = 12, answer choice A is not correct.

B) 13

Since 12 + 1 = 13, answer choice B is not correct.

C) 14

Since we cannot get x + y to sum to 14, answer choice C is correct.

Answer: C

naturalthing · Answer

While solving the Number Properties Manhattan book, I came across this question.

When positive integer x is divided by 5, the remainder is 2. When positive integer y is divided by 4, the remainder is 1. Which of the following values CANNOT be the sum x+y?
A 12
B 13
C 14
D 16
E 21

Now the question is fairly straightforward. My doubt is regarding how come we are eliminating answer choice B i.e. 13. It is a possible value of x + y + 3 (remainder) as it can be 5 x 2 + 4 x 0 + 3, but the question has defined x and y to be positive integers only i.e. whole numbers greater than 0.

Hope by doubt is clear - Can someone explain to me how they eliminated Answer Choice B.

aviral101032 · Answer

Hi, When you consider possible cases for x you can assume values to be 2,7,12,17 etc and when you consider the same case for y you can assume values to be 1,5,9,13,17 etc. Now when you take x=12 and y=1 you get the value as 13. Hope this helps

tejasvkalra · Answer

Is there any easier way of doing this ? I got the right answer in under 2.5 mins but still felt that the method was sloppy

SVaidyaraman · Answer

x gives remainder 2 when divided by 5 i.e., x = 2,7,12....
y gives remainder 1 when divided by 4, ie., y= 1,5,9...
only 14 cannot be the sum of x +y

woohoo921 · Answer

ScottTargetTestPrep

Hi Scott,

I do not understand how x can equal 2 and y can equal 1. 2/5=0.4 --> there is no remainder of 2. 1/4=0.25-- > there is no remainder of 1.

What am I missing?

Thank you!

ScottTargetTestPrep · Answer

2 divided by 5 has a quotient of zero, and a remainder of 2.

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.

When positive integer x is divided by 5, the remainder is 2.

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

When positive integer x is divided by 5, the remainder is 2.

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