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15 Dec 2023, 07:01
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C
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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.
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.
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16 Dec 2023, 13:46
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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.
General Discussion
15 Dec 2023, 07:57
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Bunuel
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
