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 
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For other subjects:
 ALL YOU NEED FOR QUANT ! ! ! 
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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.

vasu1104 · Answer

x = 5Q + 2
y = 4P +1

now for x, values could be= 2,7,12,,17,22
for y, value could be= 1,5,913,17

A 12= 7+5
B 13= 12+1
C 14 not possible 
D 16= 7+9
E 21 = 12+9

choice C

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

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

Prep Toolkit

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