12 Days of Christmas GMAT Competition - Day 5: When positive integer x

Question

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When positive integer x is divided by 11, the remainder is m and when positive integer y is divided by 11, the remainder is n. What is the value of m + n ?

(1) x + y is a multiple of 11.
(2) x - y is a multiple of 11.

missionmba2025 · Answer

x = 11p + m

0 \leq m \leq 10

y = 11q + n

0 \leq n \leq 10

(1) x + y is a multiple of 11.

x + y = 11(p+q) + (m+n)

0 \leq m+n \leq 20

Hence, m + n can equal to be 0 or 11. Not sufficient.

(2) x - y is a multiple of 11.

x - y = 11(p-q) + (m-n)

One possibility is when m = n = 0.

We can have other possibility is when x = 12, y = 1. In this case m + n = 2

So not sufficient.

Statements (1) + (2)

Combined we can have only one possibility. m + n = 0.

IMO C

vishal_gupta · Answer

Let x=11a+m and y=11b+n S1:- x+y = 11c => 11a+m+11b+n = 11c => 11(a+b) + (m+n) = 11c For RHS to be a multiple of 11 LHS should also be a multiple of 11. Since 11(a+b) is already a multiple of 11, therefore m+n should also be a multiple of 11 and since m,n<11, their sum<22. So either m+n=0 or 11. Not sufficient. S2:- x-y = 11d => 11a+m-11b-n=11d => 11(a-b)+(m-n)=11d m,n<11 therefore (m-n)max = 10 (m=10 and n=0). Only possible multiple of 11 is 0 for (m-n) when either m=n=0 or m=n=any number. Not sufficient. S1&S2:- We get m+n=0. C is correct.

Catman · Answer

Let x = 11a+m and y = 11b+n , Since x and y are positive integers, m and n are non-negative integers and always less than divisor (11)
 0\leq{m}<{11}  and  0\leq{n}<{11}

0\leq{m+n}<{22}

(1) x + y is a multiple of 11. 
x+y = 11(a+b)+ (m+n)
For x+y to be divisible by 11, either m+n =0 or multiple of 11 i:e 11

Case 1: when x=22 and y=33, m=0, n=0 , m+n=0
m+n=0 ( in which both m and n are 0)

Case 2: when x= 18 and y =15, m =7 and n=4, m+n=11
m+n =11
 Insufficient

(2) x - y is a multiple of 11. 
x-y = 11(a-b)+ (m-n)
For x-y to be divisible by 11 either m-n =0 or multiple of 11
Case 1: When m-n =0 or m=n 
A. x=14, y= 25, m=3, n=3, m-n=0
m+n=6

B. x=15, y= 26 m=4, n=4, m-n=0
m+n= 8 
Multiple sum of values of m+n exists.

Insufficient

Combining 1 and 2 
We get m-n=0 and m+n=0 or m+n=11. Since m and n are integer  m+n
eq{11} 
m+n=0 and m-n=0 
This is only possible when m=n=0. m+n=0

IMO C.

Bunuel · Answer

GMAT Club Official Explanation:

When positive integer x is divided by 11, the remainder is m and when positive integer y is divided by 11, the remainder is n. What is the value of m + n ?

(1) x + y is a multiple of 11.

If x = 1 and y = 10, then m = 1 and n = 10, giving m + n = 11. However, if both x and y are multiples of 11, then m = n = 0, resulting in m + n = 0. Not sufficient.

(2) x - y is a multiple of 11.

If x = 12 and y = 1, then m = 1 and n = 1, giving m + n = 2. However, if both x and y are multiples of 11, then m = n = 0, resulting in m + n = 0. Not sufficient.

(1)+(2) Summing the equations from the statements above, we have:

(x + y) + (x - y) = (a multiple of 11) + (a multiple of 11)

2x = (a multiple of 11)

Consequently, x must be a multiple of 11. Since x + y is also a multiple of 11, y must be a multiple of 11 as well. Therefore, both m and n are 0, making m + n = 0. Sufficient.

Answer: C.

12 Days of Christmas GMAT Competition - Day 5: When positive integer x

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